NELSON'S i 'OMMON-8CHOOL ARITHMETIC: THE LOW I THE METEJ I if In- ME As LISH IN MEMORIAM 111 FLOR1AN CAJORI NELSON'S; COMMON-SCHOOL ARITHMETIC: /> DESIGNED FOE, THE USE OF THE LOWEST AS WELL AS THE HIGHEST CLASSES, AND CONTAINING THE APPLICATION OF ARITHMETIC TO THE GENERAL PURPOSES OF LIFE, AND THE METRIC SYSTEM OF WEIGHTS AND MEASURES, KECENTLY ADOPTKT) BY CONGRESS. Bv RICHARD NELSON, PRESIDENT OF NELSOX'S VNTON BlTSTNESS COLLEGE, AND AUTHOR OP NELSON'S MERCANTILE ARITHMETIC. SECOIT3D lEZDITIOIbT. u CINCINNATI: R. W. CARROLL & CO., PUBLISHERS, No. 117 WEST FOURTH STREET. 18G7. \ Entered according to Act of Congress, in the year 1867, by KIC1IARD NELSON, In the Clerk's Office of the District Court of the United States, for the Souther* District of Ohio. PREFACE. THIS treatise differs so materially from others of ita class, that the space of a preface will not suffice to give reasons for the changes made. The author will, therefore, have to content himself with stating that, in the^lassifi^ catjoj] and treatment of subjects, he has been guided by his own experience and that of the most distinguished educators of the country ; and in the application of the science to the purposes of life, his authorities have been practical men, who were familiar with the defects of the old system, and desirous that the rising generation should be trained with more direct reference to their probable future callings. improvements in machinery having almost super- ^ seded the necessity for a knowledge of arithmetic in mechanic arts, little has been done to adapt the treatise to such purpose; hence, it partakes largely of the mercan- tile character. This, the author considers, will be its highest recom- mendation, as the destination of most American youth ia business; and, especially, as every man in this great Na- tion of Commerce is more or less engaged in mercantile pursuits. (iii) iy PREFACE. The author acknowledges numerous acts of kindness and courtesy received, in the preparation of this work, from officers of the Government, gentlemen of the legal and mercantile professions, teachers in different parts of the country, especially from the Principals of the Pub- lic Schools of Cincinnati, most of whom generously and unreservedly tendered advice regarding topics of interest to their schools. To the distinguished educators, A. J. lliCKOFP and JOHN HANCOCK the latter the friend and associate of the author he is under peculiar obligations. The former gentleman, having been the first to adopt his Mercantile Arithmetic, was the most competent to suggest further improvements and hints for the adaptation of arith^ metic to common school purposes; while the latter, from his long experience as Principal of the First Intermediate School in this city, and his immediate knowledge of the wants of teachers, proved a valuable companion during the progress of the work, aiding constantly in the revision of the proof-sheets and tendering professional advice. Though little reliance has been placed on written au- thority, many works have been called into requisition, principally of a legal, scientific and educational character. Of these, free use has been made, especially in the de- partment of the tables, which, it is hoped, will be found to subserve the present wants of every-day life. liicHARD NELSON. CINCINNATI, August, 1SOG. A VOCABULARY OF TECHNICAL TERMS USED IN BUSINESS. ABATEMENT, or deduction, an amount taken off a bill for prompt payment, damages, etc. ACCEPTANCE, agreeing to price or terms proposed; a bill with one'8 name written in such a way as to bind for payment. See page 197. ACCOMMODATION PAPER, a bill or note used to raise money, and not to pay a debt. ACCOUNT, detailed statement of goods sold. A statement showing the indebt- edness of one person to another. ACCOUNTANT, a professional calculator ; one skilled in book-keeping. ACCOUNT-BOOK, a ruled book in which accounts are kept, ACCOUNT CURRENT, a plain statement of a running account between two persons. See page 194. ACCOUNT SALES, a detailed statement of goods sold, made by an agent to his principal. ACQUITTANCE, a written discharge ; a receipt in full for money due. AD VALOREM, according to value, an assessment for custom duty. ADVANCE, a sum of money paid before value is received. ADVENTURE, a doubtful speculation ; a term used in book-keeping for goods shipped to be sold on commission. ADVICE, mercantile intelligence. AFFIDAVIT, a declaration in writing, made on oath before a magistrate, etc. AGENT, one who acts for another. ANNUITY, a sum of money paid periodically. ANNUL, to make void ; to cancel. ANTEDATE, to date beforehand. APOTHECARIES' WEIGHT, the weight used in compounding medicines. APPRAISER, a valuator. ARBITRATION, reference of a controversy or dispute to persons chosen by the parties. ASSESSOR, or Surveyor, one whose duty it is to estimate the value of property for taxation. ASSETS, the funds and property of a trader or person in business. ASSIGNEE, one to whom an assignment is made. 5 G NELSON'S COMMON-SCHOOL ARITHMETIC. ASSIGNMENT a conditional transfer of property, making it over for safe keep- ing. ASSIGNOR, one who makes an assignment. ASSURANCE. See Insurance. AVERAGE, allowance made for loss at sea; a rule in arithmetic. AUDIT, to examine books, vouchers, etc. AUDITOR, one who inspects or examines and certifies accounts. B. BAGGAGE, the wearing apparel, trunks, etc., of a traveler. BALANCE, a well-known instrument for weighing ; to find the difference be- tween two sides of an account', also that difference. BALANCE OF TRADE, the difference of the money value of the produce re- ceived and exported. If a country receives more produce and manufac- tures than it ships, the balance of trade is said to be against it. BALANCE-SHEET, a paper containing a concise statement of a merchant's accounts. BALANCING BOOKS, the business of making a balance-sheet from the ac- counts in the ledger. BALE, a package of goods or produce. BANCO, a word used as a prefix to paper money of some parts of Europe. BANK-BOOK, the pass-book of a bank. BANKER, a dealer in money. BANK HOURS, from 9 to 3 o'clock. BANK NOTE, a bank-bill payable to bearer. BANKRUPT, one who is not able to pay his debts. BANKS. See page 154, BANK STOCK, the shares of a banking company. BEAR, a term used to designate a person who makes it his business to depress the price of stocks, in order to buy up. BILL OF ENTRY, a list of goods entered at the custom-house. BILL OF EXCHANGE. See page 190. BILL OF LADING, a receipt from a railroad, ship, etc., for goods entered for conveyance from one place to another. BILL-HEAD, a printed form, with name of business or address. BILL, or BILL OF PARCELS, a detailed account of goods sold. BILL OF SALE, a contract under seal for the sale of goods. BILLS PAYABLE, the name given by a merchant or other person to notes made and issued, or bills, drafts, etc., accepted by him. BILLS RECEIVABLE, all notes taken or given in payment, except one's own. BLANK CREDIT, permission given by house or person to draw money on ac- count. BON A FIDE, in good faith. BOND, a note or deed given with pecuniary security. BONDED GOODS, those for which bonds are given for the duties instead of money. BROKER, an agent or factor. See Broker, page 95. BROKERAGE, the percentage, commission, etc., paid to a broker for buying or selling. VOCABULARY OF TECHNICAL TERMS. 7 BULL, a term applied to a broker or stock jobber, who interests himself to raise the price of stocks in the market, in order to command a high sum for those he holds. BULLION, uncoined gold and silver. C. CAPITAL, stock in trade ; the amount of assets employed by a person or com- pany in business. CAPITALIST, a man of large property or means ; one who has large sums in- vested in stocks. CAPITATION, a poll tax; a tax levied on male adults. CARGO, a ship's load. CARRIAGE, the charge made for conveying goods from one place to another. CARTAGE, the charge for carrying goods on a cart. CASE, a box for holding goods or merchandise. CASH, the general name for coin and bank notes ; sometimes checks and sight bills of exchange are called cash. CASH-BOOK, the book in which merchants and others enter the money paid out and taken in. CASH CREDIT, the privilege of drawing money at a bank, obtained by depos- iting suitable security. CASHIER, one who has charge of money. CELLARAGE, privileged charge of rooms underground. CERTIFICATE, testimony given in writing; a paper granting some particular privilege. CHAMBER OF COMMERCE, an association of merchants for the protection of trade. CHARTER, license from government to pursue certain kinds of business. CHARTER PARTY, a contract in writing between the owner and freighter of a vessel. CHATTELS, all goods and real or personal property, except real estate. CHECK, an order on a bank for payment on demand. CHECK-BOOK, a printed book of blank checks. CHEST, a box or package; tea and opium are packed in chests. A chest of opium contains 111% Ibs. ; tare allowed, \% Ibs. A chest of tea is variable. CIRCULAR, a printed letter of advertisement. CLOSING AN ACCOUNT, balancing the two sides, by placing the difference on the smaller side, under the name of balance or profit and loss, and drawing lines beneath. CLERK, an assistant in a store, office, etc. See page 198. COLLECTOR, one authorized to receive money for another. COMMERCE, the business of exchanging one commodity for another; buying and selling; mercantile business. COMMERCIAL, pertaining to commerce. COMMISSION. See Commission, page 139. COMPANY, a number of persons associated in business. COMPENSATION, remuneration or reward for injury or services. COMPETITION, rivalry, contention for contract or supremacy. CONSIDERATION, a bonus; a sum given on account for any thing. CONSIGN, to send goods to an agent or factor for sale. 8 NELSON'S COMMON-SCHOOL ARITHMETIC. CONSIGNEE, a person \vho receives goods on trust, or to be sold on com* mission. CONSIGNMENT, goods sent to a distance for sale by an agent. CONSIGNOR, the person or party who consigns. CONSOLS, public stocks in England. CONSUMER, one who uses or expends goods. CONSUMPTION, a using up; the quantity consumed. CONTINGENT, a share arising from an adventure; doutbful. CONTRA, on the other side ; per contra, a writing on the opposite side. CONTRABAND GOODS, articles on which there are heavy duties, or those wholly prohibited by government. CONTRACT, an agreement between parties for a lawful consideration ; a bargain. CONTRACTOR, one who bargains. CONTRIBUTION, a joint payment of money to an undertaking. CONVEYANCE, a legal document, transferring land or other property from one person to another; the transport of goods or passengers from one placo to another ; a vehicle. COOPERAGE, money paid to a cooper, or charges made for repairs of casks, etc. COPARTNER a person engaged in partnership. COPYING INK, adhesive ink, prepared with gum, etc., for transferring writing. COPYING PRESS, an instrument for taking impressions from writing; copy- ing letters, etc. CORRESPONDENT, a letter writer ; news writer for a periodical. COUNTER ENTRY, a contrary entry. COUNTERMAND, a contrary order. COUNTING-HOUSE, or COUNTING-ROOM, a merchant's office. COUPON, that part of a bond or other instrument designed to be cut off. CREDIT, giving trust; goods supplied without present payment. CURRENCY, paper money and coin established as the circulating medium of a country. CUSTOM, a tax levied on goods imported or exported. CUSTOMER, a regular buyer of goods at a stated place. CUSTOM-HOUSE, the place appointed to receive custom. CUSTOM-HOUSE ENTRY, a statement made, and fees and expenses paid in clearing a ship. D. DAMAGE, injury inflicted or sustained. DAMAGED GOODS, articles of merchandise or produce which have been in- jured. DAY-BOOK, the book in which merchants record daily transactions. DAYS OF GRACE. See page 142. DEBIT, to make any thing debtor in one's books ; a charge entered DEBIT SIDE, the left side of a page in the ledger. DEBT, something due to another. DEBTOR, one who owes another. DECIMAL CURRENCY, moneys reckoned by tens, as the United States cur- rency. DEED, a legal instrument of agreement under seal. VOCABULARY OF TECHNICAL TERMS. 9 DEFAULT, a failure of payment. DEFAULTER, one who makes away with the public funds intrusted to his en re. DEFICIT, a deficiency; something wanted DEPOSIT, a lodgment; a pledge or pawn; money intrusted to the care of others, DEPOSITOR, one who has money lodged in bank for safe keeping. DEPRECIATION, lessening in value. DESPATCH; to transmit, to forward goods, papers, etc. See Dispatch. DETERIORATION, damage done ; wear and tear sustained. DIRECTOR, a manager, a superintendent selected by a company or board. DIRECTORY, an alphabetical guide or address book to the inhabitants of a city. DISCOUNT, a deduction ; something thrown off the amount of a bill or note; the sum paid by way of interest for the advance of money at bank. DISCOUNT BROKER, one who loans money on notes of hand. DISCOUNT DAY, some banks discount only on stated days, called discount days. DISPATCH, a letter or message by telegraph. DISSOLUTION breaking up of a copartnership. DITTO, the same. DIVIDEND, interests on stocks ; a share of the proceeds of a joint stock spec- ulation. DOCK, a secure landing for ships ; a place for landing cargoes ; also a place to build or repair ships. DOUBLE ENTRY, a method of keeping books, which considers every busi- ness transaction contains both a debit and a credit. DRAFT, an order to pay money; a deduction from the weight of goods; a rough copy of a writing, etc. ORAW, to write an order on another for money or goods. DRAWEE, the person on whom the bill is drawn. DRAWER, the person who draws a bill. DRAYAGE, the charge made for goods carried on a dray. DRUGGIST, one who sells drugs, chemicals, paints, etc. DRY GOODS, a commercial name for cottons, woolens, laces, etc. In England, for grain, coal, etc. DUPLICATE, a copy ; a second article of a kind. DUTY, a tax on goods or merchandise. E. EFFECTS, goods, property on hand, one's possessions. ENDORSE, to employ to the exclusion of every thing else. See Indorse. ENTERPRISE, an adventure ; a projected scheme. ENTRY, a record made in a business book ; depositing a ship's papers ofc. landing. ENGROSS, to monopolize. ESTIMATE, to appraise or value ; to judge by inspection. EXCHANGE, giving one commodity for another; a place of meeting; per- centage arising from the sale of bills, etc. EXECUTOR, a person appointed to carry out the intentions of a testator. EXHIBIT, a voucher or document produced in a court of law. 10 NELSON'S COMMON-SCHOOL ARITHMETIC. EXPENDITURE, a charge or disbursement; outlay for expenses. EXPORTER, a shipper who sends goods or produce to another country for sale. EXPORTS, goods sent out of a country. EXPRESS, a special messenger; a species of conveyance. F. FACE, the amount for which a note is drawn ; also the side on which the writ- ing is made. FAC-SIMILE, an exact copy. FACTOR, an agent or broker. FAILURE, a term for suspension of payment; breaking up of business. FANCY GOODS, ribbons, silks, satins, etc. FEE, a gratuity ; the charge of a professional man for services. FEE SIMPLE, a property acquired by inheritance or that owned without con- ditions. FELLOWSHIP, companionship, partnership. FINANCE, ready money, funds or resources. FINANCIER, one skilled in money matters. FIRE INSURANCE, security against loss from fire, obtained by the payment of a small fee. FIRE POLICY, the document received from an insurance house when goods are insured. FIRM, a copartnership, a house of business. FLAT, low, dull, inactive. FLUSH, full ; an abundance of money. FORESTALL, to buy up goods or produce before the regular time of sale. FOLIO, page. FORWARDER, an agent who attends to the conveyance of goods, etc. FORWARDING HOUSE, merchants who forward goods from one place to an- other. FREIGHT, a load ; charge made for carrying goods on ship or railroad. FUNDS, ready money. See Public Funds. FURS, preserved skins of wild animals, with fine thick hair. FUR TRADE, the business of dealing in furs. G. GAUGE, to measure the contents of vessels, or barrels, casks, etc. ; a measure or standard. GOODS, a general name for movables. GROCER, a dealer in sugar, spices, dried fruits and articles of food for the table. GROSS, the whole weight of merchandise and box, barrel, etc. ; 12 dozen; a great gross is 12 times 12 dozen. GUARANTEE, or WARRANTY, indemnity against loss; one who binds him self to see the stipulation of another performed. GUNNY BAGS, coarse sacking made in India, used for holding coffee, rice, etc. H. HAND, a measure of four inches, used for taking the height of horses. HARDWARE, goods manufactured from iron. HAWKER, a peddler. HOGSHEAD, a large cask, formerly a measure of capacity. HORSE REPOSITORY, a place kept for the sale of horses. VOCABULARY OF TECHNICAL TERMS. 11 HUNDRED WEIGHT, a hundred pounds. In England, 112 pounds HONOR, to accept a draft, by paying or promising to pay. HYPOTHECATE, to pledge as security. I. IMMOVABLES, lands, houses, fixtures, etc. IMMUNITY, freedom from tax, office or obligation. IMPERATIVE, positive, commanding. IMPERISHABLE, not subject to decay or waste. IMPORTED, brought from a foreign country. IMPORTER, one who brings goods from abroad. INCOME, receipts, gains from labor, trade, etc. INCONVERTIBLE, not transmissible ; ftmds that can not be converted stocks. INDORSE, to write one's name on the back of a note or draft. INDORSEMENT, a writing on the back of a note or other paper INDORSER, one who makes an indorsement. INITIALS, the first or capital letters of a name. INLAND BILLS. See page 196. INTEREST, right or share in business. See page 142. INSOLVENT, want of ability to pay. INSURANCE. See page 244. INTELLIGENCE-OFFICE, a registry office for domestics looking for situa- tions. INVENTORY, a list of goods or effects. INVESTMENT, capital employed ; money at interest. INVOICE. See page 100. INVOICE BOOK, a book containing invoices or copies of invoices. J. JOINT-STOCK COMPANY, an association of men to carry on heavy under- takings. JOURNAL, an intermediate book between a day-book and a ledger. L. LAND-WARRANT, a title to a lot of public land. LEASE, a deed ; a contract for the use of property. LEGAL TENDER, the authorized coins or money of a country. LETTER OF ADVICE, intelligence. LETTER OF ATTORNEY, legal authority to act for another. LETTER OF CREDIT, a letter from a mercantile or banking-house given to a traveler, by which he can collect money in a foreign country. LEVEE, shipping place or landing. LICENSE, a grant. LIEN, a legal claim ; power to prevent sale by another. LIGHTERAGE, charges for conveying goods by a lighter. LIQUIDATION, the act of settling debts. LIVE STOCK, animals kept on a farm or for sale, as cows, horses, hogs, etc. LLOYDS, an establishment in London for the classification of ships; a place of assembly for merchants and underwriters to assemble. LUGGAGE, baggage, the clothing, etc., of a traveler. 12 NELSON'S COMMON-SCHOOL ARITHMETIC. M. MANIFEST, a list or exhibit of a vessel's cargo. MARINE, belonging or pertaining to the sea. MARKET, a place of sale ; price. MARKETABLE, what may be readily sold. MART, a market. MATERIALS, the substances from which goods and wares are made up. MATURITY, the time when a bill or note falls due. MEASUREMENT' GOODS, light goods taken on freight by measurement. MERCHANDISE, trade goods or wares ; goods bought to sell. MINT, an official place for coining money. MONEY BROKER, a dealer in money, bills of exchange, etc. MORTGAGE, a pledge of land for the payment of a debt. MORTGAGEE, the person who holds the pledge. MORTGAGER, the person who gives the pledge. MOVABLES, things that can be moved easily, as furniture, etc. N. NEST, a set of tubs, buckets, baskets, etc. NET, the clear amount, the quantity remaining after all deductions. NET PROCEEDS, the remainder after deducting expenses. NOTARIAL SEAL, the seal of a notary public. NOTARY PUBLIC, an officer authorized to attest documents and protest bills of exchange, notes, etc., for non-payment or non-acceptance. NOTE, a written promise to pay a debt ; a memorandum. O. OBLIGATION, a bond, a binding agreement. ORDER, a request to pay ; commission given to supply goods. ORDER-BOOK, a manufacturer's book, in which orders are copied. P. PACKAGE, a bundle. PACKER, a person who receives goods to pack for shipment. PANIC, a monetary pressure or crisis. PAPER, an article in common use; the name given by merchants to notes, bills, etc. PAPER CURRENCY, paper money of a country. PARCEL, a small package or bundle. PARTNER, an associate ; the member of a copartnership. PAR OF EXCHANGE, the value of money, both in weight and fineness, when compared with that of other countries. PASS-BOOK, a small book kept between a bank and its depositors, a merchant and his customers. PAYEE, the person to whom money is to be paid. PEDDLE, to carry about goods for sale. PERSONAL PROPERTY, money and movable goods outside of one's business. PETTY CASH-BOOK, a memorandum book of small receipts and expenses. POLICY, a writing of agreement given by insurance companies. POST-DATE, to date after the real time. POSTING, transferring from day-book, journal, etc., to the ledger. VOCABULARY OF TECHNICAL TERMS. 13 POWER OF ATTORNEY, authority to act for another. PRICE CURRENT, a published list of market prices. PRIME, superior. PRINCIPAL, the hear! of a school or business. PRO-FORMA, according to form. PROMISSORY NOTE, an engagement in writing to pay a specified sum at a a stated time. PROSPECTUS, outline or sketch of an institution, business, book, etc. PROTEST, an official notice from a notary public of the non-payment of a bill, preparatory to legal proceedings. PURVEYOR, one who supplies provisions. Q- QUARTER, the fourth part of any thing; a measure of weight, 25 Ibs. ; also a measure of length, 9 inches. QUOTATIONS, current price for stock and shares, or articles of produce in the market. R. REBATE, discount, a reduction. RECEIPT, an acquittance, acknowledgement of payment. RECEIVER, a cashier, a person appointed to take charge of property in litiga- tion. RECEIVING HOUSE, a depot or store. RESOURCES, funds, assets, that which may be converted into supplies. RETURNS, profits or receipts in business ; accounts of goods sold by an agent. REMITTANCE, bills or money sent from one house to another. RENEWAL of a bill or note, giving a new note for a longer time ; extension of time on notes, etc. S. SALE, an auction ; the disposal of goods to a private bidder. SALVAGE, a reward claimed for saving property from loss at sea. SAVINGS BANKS, banks of deposit, where interest is allowed on the amount lodged SCHEDULE, an inventory of goods on parchment or paper. SCRIP, a receipt or acknowledgment for installments paid on stocks; a partial receipt, to be substituted by a receipt in full when all has been paid. SECRETARY, a head clerk or writer ; the recording officer of a society. SHIP-LETTER, a letter forwarded by private ship, instead of a packet char- tered for that purpose. SHIPPED, transmitted by sea; goods forwarded by any conveyance. SHIPPING CLERK, a person who attends to shipping of goods. SHIPMENT, the goods forwarded by railroad or steamboat; a term in double- entry book-keeping. SHOP, a work-room ; the name given to stores in England. SIGHT, or AT SIGHT, the time when a bill is presented to a person on whom it is drawn. SIGNATURE, the name of a person written by himself. SILENT PARTNER, a partner who puts in capital, but does not take an active pjirt in the business. SLEEPING PARTNER, the term used in Britain for silent partner. 14 NELSON'S COMMON-SCHOOL ARITHMETIC. SMUGGLING, passing goods into a country clandestinely, so as to avoid the duties. STAPLE, the commodities which always meet with ready sale; the principal articles of produce or manufacture of a country. STERLING, according to a fixed standard ; a term applied to the money of Great Britain. STOCK, goods kept for sale ; materials of manufacture ; animals on a farm. STORAGE, charge for the use of a warehouse. STORES, supplies laid in for a ship. SUNDRIES, in hook-keeping, more than one; plurality. SUSPENDED, temporarily removed from employment; alleged inability to pay debts ; stoppage of work or business. T. TELLER, an officer in a bank who receives or pays money. TIERCE, a cask containing about 42 gallons. TRADE, the commerce of a country ; to exchange commodities; a bargain. TRADE ALLOWANCE, trade price; a discount allowed to merchants. TRADESMAN, a mechanic; in England, a storekeeper or retailer. TRAFFIC, trade, exchanging commodities. TRANSCRIPT, a copy. TRANSFER, a change of property, government funds, etc. TRANSHIPMENT, the act of removing from one ship to another. TRANSPORTATION, the conveyance of goods ; a name for a forwarding com- pany. TRANSITU, on passage ; on the way from one place to another. V. VENDOR, a seller; one who disposes of goods or property. VENDUE, a public sale ; an auction. VOUCHER, an instrument of writing ; a document produced to substantiate a statement of disbursements. W. WALL STREET, the street in New York City where tne principal bankers are located. WAREHOUSE, store-room ; a place for depositing goods. WRIT, an official notice from a law court. To TEACHERS?. Attention is requested to the arrangement of the answers, which makes the book subserve the purpose of one without answers and a key. In some places, as on page 44, the sum of the answers to a group of questions is given. Should the pupil construe his work to suit this answer, the teacher can detect the fraud by working auy one of the questions of the group. In other places, as on page 40, the answers are arranged promiscuously, with usually one or more than the number of questions. This prevents copying or working from the answer, and yet encourages the learner to study. MONEY, WEIGHTS AND MEASURES. FEDERAL MONEY consists of four kinds : gold, silver, nickel and paper. The smallest gold coin is of the de- nomination of one dollar. Other gold coins are the quar- ter-eagle, half-eagle, eagle and double-eagle. The silver coins are the dollar, half-dollar, quarter-dol- lar, dime, half-dime and three-cent piece. These are also represented by paper of the same de- nomination. The nickel coins are the one and two-cent pieces. The established currency of the United States consists of the eagle, dollar, dime and mill; but accounts are kept in dollars and cents only. TROY WEIGHT is used in the sale of gold and silver and at the mint for coinage: 24 grains 1 pennyweight; 20 pennyweights=l ounce ; 12 ounces=l pound. The signs are gr., pwt., oz. The caret, when applied to gold, is only a comparative weight, used to indicate the proportions of pure gold and alloy. It is ^ part of the mass of whatever weight. 18 carets fine is J| gold, or 18 parts gold and 6 alloy. COMMERCIAL WEIGHT, used in selling groceries, drugs, etc.: 16 ounces=l pound; 2000 pounds 1 ton. Signs, OZ., Us., T. NOTE. The ounce and pound are the principal parts of avoir- dupois weight in use in the United States. Iron ore and hemp are (15) 16 KELSON'S COMMON-SCHOOL ARITHMETIC. weighed by the old standard, 112 pounds to the hundred (civt.) and 20 hundreds, or 2240 pounds, to the ton. See Weight of a Toil page 17. MEASURES OF CAPACITY. The units of measurement are the gallon for liquid, and the bushel for dry measure. The gallon contains 58872.2 grains Troy of the standard pound of distilled water at 39 F., weighed in air of the temperature of 62, and barometer pressure 30 inches. It contains nearly 231 cubic inches. The bushel contains 543391.89 grains Troy of distilled water, under the above conditions, and is thus the Win- chester bushel of 2150.42 cubic inches. DRY MEASURE, used for measuring grain, fruit, etc. : 2 pints 1 quart; 8 quarts 1 peck; 4 pecks 1 bushel. Signs, pt., qt.j ph., bu. NOTES. 1. The U. S. bushel is a cylindrical vessel, 8 inches deep and 18 diameter, inside, and contains 2150.42 cubic inches. 2. By statute in Ohio, the bushel for stone coal, coke and un- slacked lime contains 2688 cubic inches, and the measure should be 24 inches at the top, 20 inches at the bottom and 14.1 deep, and contain two bushels. 3. The bushel of New York State contains 80 pounds of pure water, or 2211.84 cubic inches. LIQUID MEASURE,* for measuring all liquids: 4 gills= 1 pint; 2 pints 1 quart; 4 quarts 1 gallon. *1. Liquid measure is the old wine measure, and has superseded that of beer and ale measure, both in the United States and Great Britain. 2. The gill is seldom used, while barrels, tierces, etc., are gauged and reckoned by ? gallons. 3. The gallon contains 231 cubic inches. 4. A pint of water weighs 1 pound. 5. The capacity of cisterns, vats, etc., is usually reckoned in bar- rels and hogsheads. 31 J gallons=l barrel; 2 barrels, or G3 gal- lons=l hogshead. MONEY, WEIGHTS AND MEASURES. 17 WEIGHTS OF PRODUCE PER BUSHEL, according to usage in Cincinnati, and as fixed by statute in Ohio: u Apples dried ?age. Stat. Ibs. Ibs. 25 25 48 48 34 60 60 20 25 30 62 50 80 70 30 32 46 56 56 70 70 8 16 60 60 51 34 50 40 33 32 56 25 Ibs. .. 2240 i Peaches dried Qsage. Stat. Ibs. Ib,'. 33 33 60 60 24 118 22 60 60 50 56 56 40 60 62 60 45 45 66 56 44 44 14 50 50 14 50 50 00 45 40 30 60 60 60 77.6274 Ibs. .. 2240 Barley , Peas Barley malt, weight of b;i ir s included green Plaster and hair Beans . . ... ... Peanuts roasted Bran Potatoes Irish Bran shorts .. sweet Broom-corn Rye... Buckwheat Rye malt, wt. of bags included . . . Coal bituminous cnnnel Salt Charcoal Seed, clover Coke timothy Castor beans flax Corn, shelled hemp in ear.... 68 arid Hair, plasteriuf orchard grass.... Hungarian grass blue o*rass wet Hominy millet I-iime slacked Malt sorghurn Ship stuff Middlings Shorts ... Oats Turnips ( )nions Wheat Onion sets Water, distilled WEIGHTS PER TON Pier Iron chill mold. . Iron ore....... Pig Iron, sand molds... Blooms ... . .. 2268 '>464 Hemp .. 2240 HAV ... . 2000 THE WEIGHT OF A Flour PINT OF ounces. . ... 14 Crnsli snorfir ounces. 17 Meal . 18 Brown sugar .... 18 Butter.., .. 15 Loaf suecar..., . 19 18 NELSON'S COMMON-SCHOOL ARITHMETIC. WEIGHT OF A CUBIC FOOT OF Ibs. I Ibs. Cast iron 450.55 Wrought iron 486.65 Steel 4S9.8 Copper 565 Lead 708.75 Brass 537.75 Tin 456 White pine 29.56 Loose earth or sand 95 Common soil 1-4 Strong soil 127 Clay 135 Coal 45 to 55 Charcoal..... 18 to 18.5 Yellow pine 33.81 White oak 35.2 Live oak 70 Salt water (sea) 64.3 Freshwater 62.5 Air 07529 Steam 03689 Clay 135 Sand 113 Cork 15 Tallow 59 Brick 119 Coke 32 Ice 58 23 cubic feet of sand, 18 of earth or 17 of clay make a ton. APOTHECARIES' FLUID MEASURE, used in compounding medicines : 60 minims 1 fluid drachm ; 8 fluid drachms= 1 fluid ounce; 16 fluid ounces=l pint; 8 pints=l gal- lon. Signs, M.j minim; /., fluid drachm; /,., fluid ounce; Q, pint; Cong., gallon. MEASURES OF TIME. Time is divided into seconds, minutes, hours, days, weeks, months, years and centuries. 60 seconds 1 minute; 60 minutes 1 hour; 24 hours 1 day; 7 days=l week; 4 weeks 1 lunar month; 12 calendar months=l year; 365 days=l common year; 366 days 1 leap-year ; 365 days 5 hours 48 minutes 49.7 sec- onds, or 365| 1 solar year. A leap-year is exactly divisible by 4, and has 29 days in February. 1860 and 1864 were leap-years. The calendar months are 1. January, 31 days. 7. July, 31 days. 2. February, 28 " 8. August, 31 " 3. March, 31 " 9. September, 30 " 4. April, 30 10. October, 31 5. May, 31 " 11. November, 30 " ti. June, 30 " 12. December. 31 MONEY, WEIGHTS AND MEASURES. 19 Commencing with January, every other month has 31 days to July, inclusive; and commencing with August, every other month has 31 days to December, inclusive. CIRCULAR MEASURE is divided into seconds, minutes and degrees. 60 seconds 1 minute; 60 minutes 1 degree; 360 degrees 1 cir- cumference. Signs, ", seconds ; ', minutes ; , de- grees. 25 31' 27", 25 deg., 31 min. 27 sec. LINEAR MEASURE. Long Measure is used for measur- ing length, breadth, depth or distance. 12 inches 1 foot; 3 feet 1 yard ; 5^ yards 1 rod, perch or pole ; 40 rods= 1 furlong; 8 furlongs, or 320 rods 1 mile. Signs, in.> inches;/!?., feet; yd., yard; ?*J., rod; fur., furlong; mi. y mile. 1. In a mile there are 63360 inches, 5280 feet, 1760 yards, 320 rods. 2. The furlong is seldom used. 3. The 12th part of an inch is called a line. 4. Cloth is measured by the yard and fractional parts of a yard. MARINE MEASURE, for measuring distances at sea: 6 feet^l fathom; 120 fathoms=l cable length: 880 fath- omsI mile, called a nautical or geographical mile ; 60 geographical or 69.77 statute miles 1 degree.* The speed of a ship at sea is measured by an instru- ment called a log-line, the knots in which correspond to the number of miles sailed per hour. Knot is therefore synonymous with mile. A ship sailing at 7 knots is mov- ing at the rate of 7 miles an hour. SQUARE OR SURFACE MEASURE. Surfaces are meas- ured by taking the length and breadth, by long measure, and multiplying them together. The length and breadth *The depth of the sea is measured by fathoms. 20 NELSON'S COMMON-SCHOOL ARITHMETIC. of the surface of a foot are 12 inches each; hence, 12 times 12144 square inches 1 square foot; 9 square feet=l square yard. Land Measure : 30 J square yards=l square rod ; 40 square rods=l rood ; 4 roods, or 10 square chains 1 acre; 640 acres=l square mile; 36 square miles=l town- ship. Signs, sq. yds.) sq. rds., R., A. 1. Feet and even inches may also be used in measuring land. 2. A square of flooring or roofing is 100 square feet. 3. A square mile is sometimes called a section. 4. A square rood contains 272 j square feet; an acre, 48560 square feet. CUBIC OR SOLID MEASURE includes three dimensions, length, breadth (or width) and thickness (or depth) mul- tiplied together. 1728 cubic inches 1 cube foot; 27 cubic feet 1 cubic yard. STONE MEASURE is applied to masonry, which is some- times paid for by the foot, but usually by the perch. 24f or 25 cubic feet=l perch ; the former for private, the latter for public contracts, as railroad or government work. Wood Measure: Wood is sold by the cord, which should measure 128 cubic feet closely piled, or 138 feet if stowed in a boat or barge. A pile of wood 8 feet long, 4 feet wide and 4 feet thick contains a cord. BRICKLAYERS' MEASURE. The common dimensions of a brick are 8 inches long, 4 inches broad and 2 inches thick. There are 21 bricks in a cubic foot of wall, in- cluding mortar. A wall 8 inches or 1 brick in thickness contains 14 bricks to the square foot of surface. A wall 12 inches or 1 J bricks in thickness contains 21 bricks to the square foot of surface. A wall 16 inches or 2 bricks in thickness contains 28 bricks to the square foot of surface. MONEY, WEIGHTS AND MEASURES. 21 A fall of T *0 of an inch in a mile will produce a current in rivers. Ice 2 inches thick will bear infantry ; 4 inches, cavalry or light guns; 6 inches, heavy field pieces. PAPER. For Printers. Sizes of paper made by ma- chinery : Double imperial, 32 by 44. Super royal, 21 by 27. Double super royal, 27 by 42. Royal, 19 by 24, 20 by 25. Double medium, 23 by 26, 24 by Medium, 18J by 23J. 37 and 25 by 38. Demy, 17 by 22. Royal and half, 25 by 29. Folio post, 16 by 21. Imperial and half, 26 by 32. Foolscap, 14 by 17. Imperial, 22 by 32. Crown, 15 by 20. A sheet folded in 2 leaves is called a folio; in 4 leaves, a quarto; in 8 leaves, an octavo, or Svo.; in 12 leaves, a duo- decimo, or 12mo.; in 18 leaves, an 18mo. ; in 24 leaves, a 24mo. Stationers. 24 sheets 1 quire; 20 quires 1 ream. Bookbinders count from 16 to 20 sheets to the quire in binding account books. Wrapping Papers are sold by the ream and bundle; some reams are short count; the long count reams contain full quires. SUNDRIES : 1 barrel of flour=196 Ibs. 3 inches=l palm. 1 barrel of pork, etc. =200 Ibs. 4 inches=l hand. 1 firkin of butter=55 Ibs. 9 inches=l span. 12 articles=l dozen. 3.28 feet=l meter. 12 dozen=l gross. 3.937 or 47 J inches=l 144 dozen==l great gross. aune. 20 articles=l score. GAS. 1.43 cubic feet of gas per hour give a light equal that of a candle; 1.96 feet equal 4 candles; 3 cubic feet equal 10 candles. 22 NELSON'S COMMON-SCHOOL ARITHMETIC. HORSE POWER, in machinery, is reckoned at 33000 Ibs. raised 1 foot in a minute; but the ordinary work of a horse is 22500 Ibs. per minute for 8 hours. STRENGTH OF A MAN. The mean effect of the power of a man, unaided by a machine, is the raising 70 Ibs. 1 foot high in a second for 10 hours a day of the power of the horse. NOTE. Two men working at a windlass, at, right angles to each other, can raise 70 Ibs. more easily than one man can 30 Ibs. A foot soldier travels 70 yards, making 90 steps, in one minute, common time. In quick time, 86 yards, making 110 steps. In double-quick, 109 yards, making 140 steps. Average weight of men, 150 Ibs. each. Five men can stand in a space of 1 square yard. A man without a load travels on a level ground 8J hours a day, at the rate of 3.7 miles an hour, or 31 J- miles a day. He can carry 111 Ibs. 11 miles in a day. A porter, going short distances and returning unloaded, can carry 135 Ibs. 7 miles a day. He can carry in a wheelbarrow 150 Ibs. 10 miles a day. HAY. 10 cubic yards of meadow hay weigh a ton. When the hay is taken out of old, or the lower part of large stacks, 8 to 9 cubic yards will make a ton. HILLS IN AN ACRE. 3 feet apart, there are 4840 hills in an acre. BRITISH MONEY, WEIGHTS AND MEASURES. In Great Britain, accounts are kept in pounds, shillings, pence and farthings. 4 farthings 1 penny; 12 pence 1 shilling; 20 shillings=l pound. Signs, (farthings are written as fractions of a penny,) d.> pence; s., shillings; , pounds. ^ MONEY, WEIGHTS AND MEASURES. 23 The coins are the copper half-penny and penny; silver , three-penny, four-penny, six-penny, shilling, half-crown and crown pieces; gold, the half-sovereign, sovereign and guinea. The value of the crown is 5 shillings; the sov- ereign, 20 shillings; the guinea, 21 shillings. THE COMMERCIAL WEIGHT is the avoirdupois, of which there are in use the ounce, pound, stone, quarter, hundred and ton. 16 draclims=l ounce; 16 ounces=l pound; 14 pounds 1 stone; 28 pounds=l quarter; 4 quarters-! hundred; 20 hundred 1 ton. Signs cZr., drachms; oz., ounces; Z6s., pounds; qrs., quarters; cwt., hundreds; T., tons. IMPERIAL MEASURES OF CAPACITY for all liquids and dry goods, such as grain, potatoes, etc. : 1 gill (gl.)=8.6648 cubic inches=5 oz. water. 4 gillsr^l pint 34,65925 cubic inches=lj Ibs. water. 2 pints (pts.) 1 quart=G9.3185 cubic inches=2 Ibs water. 4 quarts (qts.)=l gallon 277.274 cubic inches=10 Ibs water. 2 gallons (gal.) 1 peck 554.548 cubic inches=20 Ibs. water. 4 pecks (pk.)=l bushel=r2218.192 cubic inches=80 Ibs. water. 8 bushels (bu.)=l quarterzirl7745.536 cubic in.=640 Ibs. water. 4 quarters (qr.)=l chaldron. 10 quarters=l last. The largest measure for liquids is the gallon; the smallest for grain, etc., the peck. In London, a chaldron of coal contains 36 bushels. NOTE. By act of Parliament, in 1824, wine, ale and dry meas- ures were superseded by the imperial measures of capacity. Time is divided into quarterly terms not recognized in the United States. In England. Lady day, or 1st term Mar. 25 Midsummer, or 2d term. ..June 24 Michaelmas, or 3d term. ..Sept. 29 Christmas, or 4th term ...Dec. 25 In Scotland. Candlemas, or 1st term. ...Feb. 2 Whitsunday, or 2d term. ..May 15 Lammas, or 3d term Aug. 1 Martinmas, or 4th term.. .Nov. 11 Linear, square or superficial, cubic measures, etc., are the same in both countries. 24 NELSON'S COMMON-SCHOOL ARITHMETIC. Articles sold by "tale," or count: 12 articles=l dozen; 12 dozen=l gross; 5 score=:l hundred; 6 score 1 long hundred. MULTIPLICATION TABLE. IX 1- l 2X 1- 2 3X 1- 3 4X 1= 4 1X2- 2 2X 2= 4 3X 2= 6 4X 2^ 8 IX 3= 3 2X 3= 6 3X 3^ 9 4X 3= 12 IX 4= 4 2X 4 = 8 3X 4= 12 4X 4r= 16 IX 5- 6 2X 5 10 3X 5= I 5 4X r >= 20 IX 6^ 6 2X 6^ 12 3X 6= 18 4X 6^ 24 ix T:= 7 2X 7 = 14 3X 7_^ 21 4X 7=r 28 IX 8 = 8 2X 8= 16 3 X 8 = 24 4X 8= 32 IX 9- 9 2X 9= 18 3X 9 = 27 4 X 9 36 1 X 10 = 10 2X10= 20 3 X 10 30 4X10= 40 1x11- 11 2X 11 22 3X11 33 4X11 44 1X12- 12 2 X 12 = 24 3 X 12 = 36 4 X 12 = 48 5X 1- 5 6X 1 7X1- 7 8X 1 8 5 X ' 2 = 10 6X 2= 12 7X 2^ 14 8X 2r^ 16 5X 3= 15 6X 3= 18 7X 3= 21 8X 3= 24 5X 4= 20 6X 4= 24 7X 4^ 28 8X 4= 32 5 X 6 = 25 6X 5= 30 IX 5= 35 8X 5^ 40 5X 6= 30 6 X 6 = 36 ?X 6r= 42 8X 6= 48 5X 7= 35 6 X 7 = 42 7X 1= 49 8 X 7 = 56 5X 8= 40 6 X 8 = 48 ?X 8= 56 8X 8= 64 5 X 9 = 45 6 X 9 = 54 7X 9^r 63 8X 9rrr 72 5X10= 50 6X10^= 60 7 X 10 = 70 8 X 10 = 80 5 X 11 55 6X11= 66 7X11 77 8XH= 88 5 X 12 = 60 6X12= 72 7X12:= 84 8 X 12 = 96 9X 1- 9 10 x i 10 11 X 1- 11 12 X 1 12 9X 2= 18 10 X ^ 20 11 X 2= 22 12 X 2 = 24 9X 3 = 27 10 X 3 = 30 11 X 3= 33 12 X 3= 36 9 X 4 ^ 36 10 X 4= 40 11 X 4 44 12 X 4= 48 9X 5= 45 10 X 6 = 50 11 X 5= 55 12 X 6= 60 9X 6= 54 10 X 6= 60 11 X 6 = 66 12 X 6=r 72 9 X 1 = S3 10 X 7= 70 11 X 7= 77 12 X 7= 84 9X 8 = 72 10 X 8= 80 11 X S= 88 12 X 8 = 96 9X 9= 81 10 X 9= 90 11 X 9= 99 12 X 9 = 108 9X10= 90 10 X 10 = 100 11 X 10 = 110 12 X 10 = 120 9X11 99 10 X 11 = 110 11 X 11 121 12X11 =132 9 X 12 108 10 X 12 == 120 11 X 12 132 12 X 12 = 144 NELSON'S COMMON-SCHOOL ARITHMETIC. I. INTRODUCTION. 1. Arithmetic is the art or science of computing num- bers. 2. The theory of Arithmetic treats of the properties and relations of numbers. 3. The practice of Arithmetic shows the application of numbers to business, the mechanic arts, etc. 4. AEITHMETICAL SIGNS. -f-, called plus, is the sign of addition. = is the sign of equality. 5-}-4=9, is read, five plus four equals nine. , called Minus, is the sign of subtraction. 5 3=2, is read, five minus three equals two. -r- is the sign of division. 9-r-3=3, is read, nine divided Ly three equah three. . is the decimal sign. Placed at the left of a number, it represents tenths, hundredths, etc. : : : : is the sign of proportion. 3 : 4 : : 6 : 8, is read, three is to four as six is to eight. 4. 4', 4", 4'", is read, four, four prime, four second, four third. f 3 (25) 26 NELSON'S COMMON-SCHOOL ARITHMETIC. I/ is called the radical sign, or sign of square root. 1/4=2, is read the square root of four equals two. -ty ' . A figure inserted, as the 3, indicates the root to be taken; 3, the cube root; 4, the fourth root. 3 2 . A small figure written as the 2 in the margin, indi- cates that the number is to be raised to a corresponding power; 2, the second power; 3, the third power. This figure is called the index of the power. II. NOTATION AND NUMERATION. 5. Notation is the art of representing numbers by sym- bols, called figures or digits. There are ten of these figures : 0123456789 naught, one, two, three, four, five, six, seven, eight, nine. The first is also called zero, or cipher. 6. When a larger number than nine is to be repre- sented, two or more figures are used. 7. Numeration is the method of reading these figures when arranged to represent numbers. For this purpose they are usually divided into periods of three from the right. 8. The first period on the right contains units, tens and hundreds, thus : 125 bund., tens, units, which is read, one hundred and twenty-five. The second period contains units, tens and hundreds of thousands, thus : 125 , 000 thous., hund., which is read, one hundred and twenty -five thousand. \ NOTATION AND NUMERATION. 27 The third period contains units, tens, and hundreds of millions, thus : 125,000,000. mills., thous., hund., which is read, one hundred and twenty-five millions. The fourth period contains billions, the fifth trilli)ns y the sixth quadrillions, the seventh quintillions. RECAPITULATION. The first period is hundreds, the second thou- Bands, the third millions, the fourth billions, etc. To read 12376421007, we point off thus: 12,376,421,097. Here are four periods the fourth is billions. The number is 12 billions, 376 millions, 421 thousand and 97. *WRITE THE FOLLOWING IN WORDS: 1 2 3 4 15 127027 1464780 27100101 25 184194 1700700 198140197 125 710107 4001001 100001009 1125 411001 7119031 600100100 23125 100020 1000010 190999009 5 3214100967831 4191006848219 WRITE THE FOLLOWING IN FIGURES: 1. Ten. 2. Seventeen. 3. One hundred and twenty. 4. Three hundred and twenty-four. 5. One thousand and eighty. 6. One hundred and twenty thousand five hundred. 7. Three hundred and ninety-seven thousand four hun- dred and forty-four. * Exercises under articles 10, 11, and 12 may precede these. 28 NELSON'S COMMON-SCHOOL ARITHMETIC. 8. Twelve millions one hundred and twenty-five thou- sand one hundred and one. 9. One billion one hundred and one thousand and one. 10. Thirty millions eighty-five thousand one hundred and seven. 11. Seventy-six trillions five hundred and forty billions one hundred and ten millions and sixty -seven. 12. Two hundred millions. 13. One hundred trillions. 14. Seven hundred billions three thousand and seven. 15. One thousand. 16. One million. 17. One trillion one hundred thousand. 18. One billion. 19. Thirteen billions three millions seven thousand. 20. One hundred and ten millions and eighty-seven. 21. Seventy-five millions six thousand and nine. 9. FEDERAL MONEY. The name usually given to the money of the United States is Federal Money. It is reckoned by tens and hun- dreds. Though there are various kinds of gold, silver and nickel coins, money is always reckoned in dollars and cents, or dollars, cents and mills. $ is the dollar sign. c, the sign for cents. m, the sign for mills. $3456.87,5, is read, three thousand four hundred and fifty- six dollars, eighty-seven cents, five mills. Mills are written one place to the right of cents. In this book mills will sometimes be separated from cents by a comma, as above. A period is used to separate cents from dollars, the two first figures on the right being cents and tens of cents. NOTATION AND NUMERATION. 29 WHITE THE FOLLOWING IN WORDS ! 12 3 $457. $31412.875 $507.974 $25.25 $167.476 $1134.643 $364.61 $365.323 $216.132 $112.57 $4767.126 $345.815 $3146.87 $116.254 $341.664 $213.18 $67.141 $416.304 $456.45 .75 .17c $1719.97 .353 .413m $7.90 $1.273 .674m $304.02 $35.144 $100.603 $117.44 $414.124 $210.301 $32.32 .67 c .164m When there are no tens of cents, a cipher is written in the ten's place, as in 304.02, Ex. 1, which is read three hundred mid four dollars and two cents. Separate the dollars, cents and mills in the following, observing to write dollars under dollars and cents under cents : 4 5 6 7 307 c 135006 m 3171 c 24136 c M 064m 2703 m 1463 m 7314 m 32002 m 753m 12195m 21364 c 100011m 35001 m 3697 c 71836 m 1826063m 61422 c 2468 m 2173 c 116003m 71922c 14193 c 1897m 31 463 c 143684 m 21314 c 1367841 m 6897 m 900013 c 2136821 m 146001 c 18987 m 41362 c 21367m 14367 c 31464 c 13687m 98076 c 314683m 8. 3141 c, 1386 c, 19301 c, 71432 c, 7321 c, 906301 c, 2136 c, 4563 c, 10001 c, 30351 c, 9878 c, 45121 c, 31645 c, 13621 c, 21001 c, 60006 c, 213621 c, 113600 c. 30 NELSON'S COMMON-SCHOOL ARITHMETIC. 9. 21360 m, 31461 m, 14565 in, 24798 m, 46503 m, 106 m, 796543m, 45631m, 245m, 14013m, 41634m, 14563m, 21364 m, 7856 m, 24130 m, 45103 m, 1467 m, 4136 m. 10. 20135m, 314102m, 14163m, 213102m, 31453m, 12063 m, 14673 m, 24135 m, 43103 m, 156307 m, 14007 m, 34617m, 21368m, 146m, 178936m, 213641m, 14567m, 36845m, 213146m, 145163m. 11. 12314m, 2136m, 100135m, 21364m, 146345m, 213984 m, 136453 m, 1467 m, 13645 m, 14107 in, 30674 m, 1451 in, 147431 m, 14345 m, 41367 m, 1456 m, 21314 m, 16789m, 13674m, 14683m, 1413m, 31463m, 16703m, 4103 m, 163 m. 12. 1456 m, 31463 m, 21456 m, 14637 m, 2136 m, 415 in, 1467m, 23460m, 1454m, 3417m, 14567m, 14674 in, 13654m, 14345m, 41631m, 14367m, 1456m, 145634 in, 1456 m, 14567 in, 4453 m, 14639 m. 13.23145m, 12346 in, 16734m, 2146m, 1678m, 1463m, 14596m, 3045m, 163201m, 146324m, 14567m, 14631 m, 21393 m, 7894 m, 14637 m, 21364 in, 14567 m, 176301 m, 21367 m, 140601 m, 14563 m. 14. Write in columns, as before, the following: Seventy- five dollars, eighty-seven cents; thirty-three dollars, sixty- one cents, five mills; seven hundred and ninety-six dollars and sixty cents; five thousand dollars; five thousand three hundred and eighteen dollars and sixty-three cents; two hundred and fifty -six dollars, fourteen cents, four mills; one hundred dollars, sixty cents, three mills. 15. Thirty-six hundred dollars, seven cents, and five mills; eight hundred thousand dollars, forty cents, seven mills; sixty-seven dollars, eighty cents; nine hundred dol- lars, seventy-five cents, six mills; three hundred and sev- enty-six dollars, six cents, four mills; sixty-four dollars, eighteen cents, seven mills; fifty-nine dollars, six cents, three mills; eight hundred dollars, one cent, three mills. NOTATION AND NUMERATION. 31 17. One thousand five hundred dollars, sixty cents, eight mills; three hundred and fifty dollars and six mills; seventy-eight dollars, eighty cents, six mills; four hundred fifty-seven dollars, sixty-four cents, seven mills. 10. ODD NUMBERS .* The numbers 1, 3, 5, 7, and 9 are called odd numbers, and every number which has one of these figures in the unit's place, as 11, 13, 15, is also an odd number. 1. Write in columns all the odd numbers from 1 to 151, observing to keep units under units, tens under tens, and hundreds under hundreds. 2. Write in the same way all the odd numbers from 151 to 351. 3. Write in the same way all the odd numbers from 351 to 601. 4. Write in the same way all the odd numbers from 601 to 901. 11. EVEN NUMBERS. The numbers 2, 4, 6, 8 are called even numbers, and every number which has or one of these as a unit figure, is also an even number. 5. Write in columns, as before, all the even numbers from 2 to 200, inclusive. 6. Write in the same way all even numbers from 200 up to 500, inclusive. 7. Write in the same way all the even numbers from 500 to 800, inclusive. 8. Write in the same way all the even numbers from 800 to 1100, inclusive. The Teacher should follow these exercises by others, from dictation, until the scholars are taught to write any sum without hesitation. *With very young learners, these exercises should precede those on pages 26 and 27. 32 NELSON'S COMMON-SCHOOL ARITHMETIC. 12. ADDINO. In the preceding exercises the learner unconsciously added 2 every time he passed from one number to another. In the following exercises he will be required to add 3, 4, 5, etc., and unite the results in the same way. He should observe to write the figures in straight lines. 9. Commencing at 1, add 3 every time until you reach 97, thus: 1, 4, 7, 10, 13, putting the numbers under each <5ther. 10. Commencing at 1, add 4 every time till you reach 121. 11. Commencing at 1, add 5 every time till you reach 161. 12. Commencing at 2, add 6 every time till you reach 200. 13. Commencing at 1, add 7 every time till you reach 232. 14. Commencing at 1, add 8 every time till you reach 265. 15. Commencing at 1, add 9 every time till you reach 307. 16. Add 3 to all the odd numbers up to 101. For this purpose write the odd numbers on the left 'of the new numbers, thus: 1 4 3 6 17. Add, in the same way, 4, 5 and 6 up to 51. 18. Add, in the sam'e way, 7, 8 arid 9 up to 51. 19. Add 3, 4, 5 and 6 to all the ven numbers up to 50. 20. Acid 7, 8 and 9 to all even numbers up to 80. 21. Add, in the same way, 2 to the following numbers: 19, 29, 39, 49, 59, 69/79, 89, 99, and add 3, 4, 5, 6 5 8 and 9 in the same way. ADDITION. 33 22. Add 2 to the following numbers, after writing them on the left: 8, 18, 28, 38, 48, 58, 68, 78, 88, 98. Add 3, 4, 5, 6, 7, 8 and 9 in the same way. 23. Add 3, 4, 5, 6, 7, 8 and 9 to the following num- bers: 7, 17, 27, 37, 47, 57, 67, 77, 87 and 97. 24. Add 4, 5, 6, 7, 8 and 9 to 6, 16, 26, 36, 46, 56, 66, 76, 86 and 96. 25. Add 5, 6, 7, 8 and 9 to 5, 15, 25, 35, 45, 55, 65, 75, 85 and 95. 26. Add 6, 7, 8 and 9 to 4, 14, 24, 34, 44, 54, 64, 74, 84 and- 94. 27. Add 7, 8 and 9 to 3, 13, 23, 33, 43, 53, 63, 73, 83 and 93. The Teacher ought to examine his scholars on the terms, signs and principles of each rule. In this subject, on the difference be- tween notation and numeration, how many figures necessary to write one hundred, how many a thousand, etc. III. ADDITION.* 13. The method of uniting two or more numbers into one is called Addition. *The Teacher will find exercises for evening study at the end of this chapter. Beginners should not be allowed to count on their fingers or talk over the process. If drilled in the use of the catch-figure by blackboard exercises, they will not afterward resort to any of the slower methods of computation. The use of the catch-figure is in part taught in exercises, page 32 and 83. For the purpose of drill, it ought to be taught as follows : "7 and 9, the unit figure is what?'' "6." "17 and 9?" "26." "27 and 9?" "36." "Observe that when 7 and 9 are added, the unit figure is 6." "47 and 9?" "57 and 9?" "67 and 9?" Classes should be exercised in this way through all the combi- nations found iii the exercises referred to. 34 NELSON'S COMMON-SCHOOL ARITHMETIC. 14. The one number is called the sum, amount, total, or footing. 15. The sign is +, and is called phis. When placed between two numbers it indicates that they are to be added together. 3-f-2 5, is read, three plus two equals Jive. 16. In performing operations in addition, it is neces- sary to write the units, tens, hundreds, etc., of the one number under the units, tens, hundreds, etc., of the other, which arranges the figures in one straight line. 1. To add together 135, 241 and 323. 135 EXPLANATION. Here units are placed under units, teng 241 under tens and hundreds under hundreds. After arranging 323 the figures thus, we commence at the right-hand column and add 3 to 1, which makes 4, to which add 5 and we have 9. Adding the tens' column in the same way, we have 9 tens, which we write in the tens' place. Adding the hundreds' column in the same way, we have 6, which we write in the hundreds' place. Find the sum of each of the following groups : * 2 3 4 5 6 7 8 3131 131 211 1534 3143 3141 2131 223 453 765 1232 2102 5432 1036 115 100 23 1002 1413 1426 5812 3799 684 999 10 11 12 13 14 15 211 3145 4512 2131 14132 14413 1613 101 4132 1035 1027 1734 34441 143 65 1712 4241 1720 4113 41104 1233 * Answers arranged promiscuously: 377, 9788,2989, 19979, 3768, 9999, 6658, 8989, 3989, 8979, 4878, 89958. ADDITION. 35 To add 325, 42 and 178. 325 42 178 Answer, 545 EXPLANATION. 1. Placing the numbers as directed, we proceed to find the amount of the first column on the right: 8, 2 and 5 are 15; that is, 1 ten and 5 units. 2. Writing the 5 units under the units, we add the 1 ten to the tens' column. #. This one added to the 7, 4 and 2 makes 14 tens or 1 hundred and 4 tens. 4. Writing the 4 under the tens, we add the 1 hundred to the hundreds' column. 5. This 1 added to the 1 aM 3 in the hundreds' column makes 5 hundreds, which 5 we write in the hundreds' place and our work is done. 4501 17. To add 4501+3213+1007+302, we write 3213 them thus: 1007 302 An*. 9023 Add together the following numbers: 18. 3478, 3167, 4199, 7854, 3456. An*. 22154 19. 1417, 210, 61907, 216, 3184. An*. 66934 20. 7894, 2176, 7, 109, 7998. An*. 18184 21. 376 + 100+71+416+709+317. An*.. 1989 22. 1006+3009+79999+7098+17. Am. 91129 23. 316+10069+9777+307+198. An*. 20667 24. 789632+4+67+879002+876+970 is how much? Ans. 1670551 25. 98632+76398+832+97+10029+97384 is how much? An*. 283372 26. 1324+4354653+12+876+97843+68473 is how much? Am. 4523181 36 NELSON'S COMMON-SCHOOL ARITHMETIC. 27. 31465+2316532+107+3790+465321+3654563+ 107653+23650+1007+30672+503102+21063 is how much ? 28.18230+476+41034+9875+65432+5678+12090+ 9387+8276+565+13654+443^how much? Ans. Sum of 27 and 28, 7344065. 29. Add together 45679+9837+18708+7967+485+ 78963+84989+12345+7069+8090+7483+96748. TAKING TWO AND THREE FIGURES AT A TIME. To enable scholars to grasp two and three figures at a time, and carry them up as*one, they might be exercised on the blackboard in such sums as the following: 136 964 4 3 9 8 4 6 8 2 1 3 7 3 1 6 4 1 9 2 3 8 4 1 1 3 4 5 3 6 693 673 212 Such exercises ought to be of frequent occurrence and scholars encouraged to answer in concert. The answers should be given instantaneously, naming only the unit figure, as shown in the column below: After writing on the right of the first column the figures produced by pairing, the teacher may lead the class in adding, thus: 17 and 3? 30 and 1? 41 and 6? 47 and 7? 54 and 1? 65 and 6? 81 and 5? 96 and 2? 108 and 11? It will be observed that the tens produced iu forming the pairs were not named. The same course should be pursued in the class, as the learner is unconscious of making as great an effort as he really does. When the ten is omitted by mistake, attention should n be called to it by giving the full number, as 15 or 11 instead of 5 or 1. 1 The other columns should be added without the aid of the marginal figures. After thorough drill in this, the class should be taught to take three figures and even four as rapidly as oae. 3456 \ 1 1345 / L 3689 ^ 9 1563 f* 9456 \ - 3689 j D 8998 | A 1898 f 9873 ) , 1678 f 1684 > 7 7893 f ' 1453 1 1763 / 2195 9876 7897 \ 2536 / S529 \ - 1438 / f ADDITION. 37 30. Find the sum of 8934, 16749, 809, 67549, 98697, 746839, 1498, 829555, 9218967, 8347912, 968000, 74685. Total of the preceding two, 20758557. Foot up the following columns : 31 32 33 34 35 31645 3454 4213 1565 3654 98760 2136 631*4 3657 1095 36875 1364 2316 5437 9014 57893 4633 1369 3457 6789 14567 9897 9306 1234 9687 34564 7879 6039 3421 5764 46387 2164 8109 6789 1567 93178 4163 9876 1746 9139 78163 4569 6789 3456 1456 ' 64518 5496 4567 1378 2345 17514 6428 5679 5932 5432 45678 8297 3263 4567 6542 21364 9287 9457 1657 1395 7198 7928 1459 6574 3642 3165 9872 1455 5638 1365 4124 8729 9375 4932 2315 1345 9314 5976 1397 9365 3146 3162 7639 9765 3510 4165 2136 7938 3765 1096 3216 9364 3959 1456 3765 36. Add together the following numbers: 313, 2109, 6785, 2736, 798, 987, 21363, 316, 4934, 2178, 1009, 396, 298, 2753, 607, 3145, 213, 6709, 6093, 190, 2130, 2160, 716, 213, 9876, 45678, 2137, 2198, 9039, 6789, 3097, 4684, 2136, 2178, 5672, 1987, 6789. Answers promiscuously arranged: 95368, 77823, 120272, 115098, 667465, 88937, 171411. The Teacher should not permit his scholars to divide these col- umns when adding, nor should he allow them to resort to the aid of strokes or practice counting on their fingers. 38 NELSON'S COMMON-SCHOOL ARITHMETIC. 37 38 39 40 41 3286 2467 34564 46321 3614 6713 109 12345 13632 1364 3654 3178 65435 14567 5436 176 145 87654 53678 7835 3976 6178 34564 86367 4678 6345 4156 13682 85432 8793 9823 7532 75671 36457 701 6023 9890 86317 21836 9804 1367 6821 24328 17354 1306 8965 9854 98713 63542 717 8632 3821 21345 78163 2103 1034 5843 1286 82645 6397 6312 1936 78654 34685 1096 4593 7136 19876 31768 2130 3687 9876 93643 65314 3107 5006 2863 6356 68231 167 7164 123 78397 64037 2109 1763 7436 21602 34685 3678 2139 1567 71346 35962 2176 8236 2563 28653 21363 5432 7860 8432 17648 78636 2137 3613 1345 82351 19854 28639 109 8736 21368 80145 1765 1756 8654 78631 87654 371 6386 1263 17639 12345 71031 9890 1345 82360 78654 1463 8243 3093 45671 12345 3168 42. Find the sum of all the odd numbers under 100. 43. Find the sum of all the even numbers under 100. 44. Find the sum of all the numbers included in Ex. 5, page 31. 45. Find the sum of all in Ex. 6. Answers promiscuously arranged: 181217, 1300099, 126362, 1325672, 136751, 143267, 52850, 2500, 2450, 10100, 2510. ADDITION. 39 46 to 49. Find the sum of all in Ex. 9, 10, 11 and 12.* 50 to 53. Find the sum of all in Ex. 13, 14, 15 and 16. 54 to 57. Find the sum of all in Ex. 17, 18, 19 and 20. 58 to 60. Find the sum of all in Ex. 21, 22 and 23. 61 to 64. Find the sum of all in Ex. 24, 25, 26 and 27. 65. Add together all the numbers from 300 to 320, in- clusive; from 3120 to 3150, inclusive; from 160 to 200, inclusive; from 1950 to 2000, inclusive. Answers in direct order: 9615, 19228, 25576, 25512, 19200, 211800. 17. To add Federal Money, we place dollars under dol- lars, cents under cents and mills under mills, and proceed as before. 66. What is the amount of the following sums of money? $32.74, $16.73, $13.09, $37.40, $16.74, $7.07. Am. $123.77. OPERATION. EXPLANATION. The sum of the first column being 27 $32.74 cents, we write the 7 and add the 2 tens of cents to the 16.73 tens' column, making 27 tens, or 2 hundreds and 7 tens. 13.09 Writing the 7, we add the 2 hundreds to the next, which 37.40 is the dollar column, and proceed as in the above. 16.74 The second column of cents might be called the dime*' 7.07 column. $123.77 Amount. 67. Find the amount of the following: $1708.25, $2076.00, $709.07, $109.88, $999.87, $370.04, $695.83, $797.00, $87.00, $400.40, $198.08, $109.65, $364.08, $217.00, $364.09, $785.66, $699.08, $776.08. *The Teacher can work most of these by Arithmetical Progres- sion. As indicated by the numbers, these exercises may be broken into three or four parts, if considered too difficult. 40 NELSON'S COMMON-SCHOOL ARITHMETIC. G8. $1670.03+81006.01+ $364.01+85432.99 $2310.00+$1068.24+$26107. 18+82136.18 $109.79+ $999.99+ $666.56+ $449.99 $777.00+$7999.00+ $6666.00+86730.15 69 70 71 72 $987.67 $716.27 $187.20 $4519.27 873.35 855.60 257.65 7864.20 473.92 219.76 330.17 9510.33 187.87 912.67 700.00 3578.84 119.16 107.30 175.85 4875.60 160.97 87.60 150.50 6115.90 176.01 101.19 37.50 9885.10 634.16 808.08 57.63 1105.75 585.26 981.61 109.87 5760.87 458.39 225.00 987.05 7901.57 385.93 811.29 1285.58 7119.85 589.38 300.92 2327.88 5006.29 107.20 10.15 8900.17 9110.11 70.99 106.30 209.18 4362.17 18.18 547.67 101.01 3210.18 1764.18 336.44 125.00 2133.64 397.27 176.33 117.45 1364.57 444.99 1275.84 361.45 2136.06 222.66 666.57 217.33 1456.27 799.88 1176.22 163.77 376.22 73. Find the amount of the following sums of money: One hundred and twenty-five dollars and twenty-five cents; Sixty-eight dollars and forty-seven cents; Three hundred and ten dollars and eighty-seven cents j Six hun- dred dollars and seven cents; Four thousand eight hun- dred and fifty dollars and eighteen cents. Answers arranged promiscuously, including those to Ex. 67 and 74: $11467.06, $64493.12, $45178062.31, $5954.84, $2624.91, $10422.81, $16802.24, $97392.79, $9457.42, $96394.79. ADDITION. 41 74. Add together the following amounts: Eighteen thousand one hundred and forty-six dollars; Seven thousand one hundred and sixteen dollars and twenty-five cents; Sixty-four thousand one hundred dol- lars and four cents; Forty-five millions and one thousand dollars; Eighty-seven thousand seven hundred dollars and two cents. 75. A merchant has 29 pieces of silk in 1 package, 35 in another, 79 in a third. In the first there are 1497 yards; in the second, 2173 yards; in the third, 4130 yards. How many pieces, how many yards? SOLUTION. 29 1497 35 2173 79 4130 Whole number of pieces 143 Whole number of yds. 7800 *76. A coal dealer sells 1254 bushels every day of the week, Sunday excepted; how many does he sell in all? To THE TEACHER. Columns of fifteen or twenty numbers may now be dictated to classes, the teacher observing to increase the speed of the scholars at every effort. The results may be called off as produced, and written by the teacher on the blackboard, or the learners may exchange slates for examination and correction. In this, as in all competitive exercises, the teacher should not wait until every member of the class has finished the work; but the tardy ones must not be overlooked, nevertheless. Means should be adopted to stimulate them to greater effort. They must be taught that they can not be allowed to fall behind without the risk of being returned to a lower class or grade. The teacher probably knows that to make boys or girls reckon rapidly he must lead; and to this end, it would be to the advantage of both teacher and pupil if such exercises as these were always impromptu. *In giving the answers, the learner should state whether it i3 bushels, pounds, etc. The abbreviation Ibs. stands for pounds. 4 42 NELSON'S COMMON-SCHOOL ARITHMETIC. 77. A merchant bought 9 bags of coffee, each bag weighing 215 Ibs., at an average cost of $21.20; what weight of coffee did he buy, and how much did he pay for it? 78. A farmer has 118 sheep, 518 hogs, 210 pair of chickens, 5 plows, 6 wagons, 1 dozen hoes, 7 horses, 10 spades and 12 pitchforks; how many animals has he, and how many agricultural implements? 79. How many pupils in a school in which there are 5 classes, the first containing 19, the second 28, the third 32, the fourth 35 and the fifth 29 pupils ? 80. Bought three boxes of oranges, in one of which there were 450, in another 469, in the last 510 oranges; how many did I buy? 81. A man walked twenty-five miles on the 20th day of the month, twenty-three on the 22d, twenty-nine on the 23d, thirty-three on the 24th day; how many miles did he walk altogether? 82. How many days in the first nine calendar months of the year ? 83. Sir Isaac Newton lived 85 years and was born in 1642; in what year did he die? Answers: 143, 1429, 273, 110, 274, 1727, 7524, 1935, 19080, 1063, 45. The Teacher should give numerous exercises besides thesd, and have his scholars work them on the blackboard before the class. HOME EXERCISES. 1. Add 1, 2, 3, 4, 5, 6, 7, 8 and 9 to 2, and take them from the result again, writing them out as below : 2 and 1 are 3; 2 from 3 leaves 1. 2 and 2 are 4; 2 from 4 leaves 2. 2 'and 3 are 5; 2 from 5 leaves 3. SUBTRACTION. 43 2. Add and subtract (take from) 3 in the same way. 3 Add and subtract 4 in the same way. 4. Add and subtract 5 in the same way. 5. Add and subtract 6 in the same way. 6. Add and subtract 7 in the same way. 7. Add and subtract 8 in the same way. 8. Add and subtract 9 in the sam^ way. IV. SUBTRACTION. 18. The process of finding the difference between two numbers is called Subtraction. 19. This difference is called the remainder or excess. 20. The sign is , and is called minus. When placed between two numbers, it shows that the one on the right is to be taken from the one on the left : 7 5=2, reads, seven minus Jive equals two. 1. To find the difference between 375 and 263. SOLUTION. 1. Writing the small number under the 375 large one, units and tens of the one under those of the 263 other, we proceed to subtract 3 from 5, which leaves 2; this we write under the 3. 112 2. 6 from 7 leaves 1 ; write it under the 6. 3. 2 from 3 leaves 1. The remainder is 112. 2. From 186436 take 165213 An*. 21223 3. From 786900 take 654300 132600 EXERCISES FOR THE BLACKBOARD. When subtracting one figure from another, the learner should be taught to see the result, rather than to reckon it or talk over the process. This can be done by such exercises as the following : Taking a row of figures as 4903781236542, point to 9, requiring the class to give the difference between it and 4 ; th,en to 0, requir- ing the difference between 10 and 9, etc. 44 NELSON'S COMMON-SCHOOL ARITHMETIC. 4. From 49368282 take 15012 Am. 49353270 5. 247896136785 8. 66145397 134286 6. 716035 15012 9. 151764824164271 7. 371150 70000 10. 37898643 276321 Total, 1113284 Total, 114645644 11. 1317^3^6^87^45 13145363435 12. 984960997610899771900986010098 13. 19899799994896 7445199821886 Total, 238677822898021. When some of the figures of the smaller number are greater than those above them, we add ten to both figures. 14. From 342 take 267. To subtract the 67 from the 42 above it, we add 10 to 10 3J. V both numbers, as indicated by the small figures; but in- stead of adding to both numbers in the same place, we 267 add 10 to the unit 2 of the upper number, and 1 (ten )to the 6 tens of the lower number. ' ^ PROCESS. 1. We can not take 7 from the 2; add 10 which makes 12; 7 from 12 leaves 5, which write. 2. Adding 1 (ten) to the 6 we have 7, which taken from 14, after adding another ten (or one hundred) leaves 7. 3. Adding 1 (hundred) to the 2 we have 3, which taken from 3 leaves nothing. The remainder is 75. 21. PROOF. By adding the remainder to the smaller number, we should get a sum equal to the larger. In the above example the remainder was 75, the smaller number 267, which added together equal the *?& larger, 342. *15. 1603845732164000000 98123456789798768 1505722275374201232 Rem. *The Teacher will require the class to read off both question and answer. SUBTRACTION 45 16. From 31642789600 take 1278899765. Rem. 30363889835. 17. From 16782466987 take 123469978. Rem,. 16658997009. 18. 216836732178491637,642178 19. 2216821783680011000999999 20. 100000000000000 1 21. 3681068213682 11194680 Total, 347044269962674. 22. From $1670 take $389.27. OPERATION. $1670.00 389.27 $1280.73 Am. When writing the following questions, be particular to To THE TEACHER. Mental exercises in adding will be found a good means of cultivating the retentive faculty for business pur- poses. Such exercises should not consist of single digits nor of fractions with terms made up of single digits, but of numbers such as the following: 27 and 64 are how much? To add these numbers the tens should be taken first: 27 and 60^87 and 4=91. $3.27 and $1.25 are how much? . 327 and 120=447 and 5=452. COMPETITIVE EXERCISES In addition might come in here, and be introduced at intervals throughout the course. A problem being written on the black- board, or dictated to the class, scholars should be required to hand in their slates in the order in which they obtain the results, when, the teacher would number them, "1, 2, 3, wrong, 4, 5," etc., calling the name of the competitor in each case and returning the slatei Such exercises should be conducted by the teacher with celerity, so that at a single glance he can tell whether the learner has the cor- rect result. Fifteen minutes will be found sufficient time to devote to such exercises at once. 46 NELSON'S COMMON-SCHOOL ARITHMETIC. arrange dollars under dollars and cents under cents. When there are no cents, write two ciphers in their place. 23. $10067.89 $2141.98 27. $60000.00 $4670.87 24. $15070.14 6160.47 28. $23678.45 4101.00 25. $1001000. 1.86 29. $1006812.00 3178.59 26. $6743147. .97 30. $678997.00 210.99 Total, $7760979.75. ' Total, $1757326.00. 31. $710356.87 $14683.29 35. $68750.37 $1416.44 32. $370968. 17987.77 36. $71000.90 87.50 33. $478979. 14780.99 37. $100000. .87 34. $100000. 374.66 38. $61987.15 .99 Total dif., $1612477.16. Total dif., $300232.62. ^39. What is the sum and difference, when added to- gether, of $36748..94 and $10968.75. 40. Borrowed of A, at different times, $146.87, $6740.18, $310.75, and have paid him $10.00, $450.18 and $61.14; how much do I owe? 41. Out of 5 hogsheads of sugar containing 5761 Ibs., I sold 3, containing 1114 Ibs., 1311 Ibs. and 1001 Ibs.; how much was left? 42. After selling 1347 Ibs. of sugar from 3 hogsheads, each containing 1000 Ibs., how much was left? EXERCISES IN TAKING THE COMPLEMENT OB "MAKING CHANGE." To THE TEACHER. Taking $1 as the complete number, require the complement of 25 c, 27 c, 30 c, 35 c, etc. It 'will be found that the complement of the teens is in the 80s, of the 20s in the 70s, of the 30s in the 60s, of the 40s in the 50s, of the 60s in the 30s, of the 70s in "the 20s, etc. "54?" Aris. 46. 35? Ans. 65. Taking $2, $3 or $5, the exercise may be practiced in the same way. *Give the denomination of the answer, whether it be dollars, pounds, etc. SUBTRACTION. 47 43. A merchant, owns goods to the amount of $3147, and lands to the amount of $2107, and is indebted $1400 : to A $200, to B $340 and to C $860; what is the amount of his net capital ? 44. A merchant sells goods for another to the amount of $4374.23, and is to receive $43.75 for his trouble, be- sides the expenses of freight, etc., which was $125.15 ; how much should he return to his principal? 45. What is the difference between 1856 and 1798? When was the individual born who died in 1857 at the age of 45 years? When will the work be completed which was commenced in 1855 and was to take eighteen years ? Answers to the last seven: 420533, 58, 1812, 2454, 1873, 1653, 7349788, 2335, 667648, 410533 .* *Give the denomination of the answer, whether it be dollars, pounds, etc. HOME EXERCISES/}- 1. 30+25=? 14. 78+65=? 27. 94+27=? 2. 27+40=? 15. 59+63=? 28. 79+86=? 3. 26+30=? 16. 72+48=? 29. 56+65=? 4. 43+27= ? 17. 37+75=? 30. 33+63=? 6. 29+12=? 18. 59+76=? 31. 98+63=? 6. 54+16=? 19. 85+64=? 32. 27+79=? 7. 33+17=? 20. 78+77=? 33. 56+65=? 8. 55+75=? 21. 99+37=? 34. 73+64=? 9. 47+43=? 22. 55+66=? 35. 55+68=? 10. 82+12=? 23. 37+74=? 36. 97+63=? 11. 95+15=? 24. 65+37=? 37. 64+67=? 12. 34+45=? 25. 59+63=? 38. 73+67=? 13. 29+64=? 26. 37+65=? 39. 45+63=? t To be written on the slate at homeland the operations recited in Bchool. 48 NELSON'S COMMON-SCHOOL ARITHMETIC. 40. 27-f-33=? 60. 59+16=? 80. 97+63=? 41. 95+64=? 61. 27+17=? 81. 84+37=? 42. 84+33=? 62. 86+59=? 82. 29+18=? 43. 27+69=? 63. 75+38=? 83. 63+97=? 44. 55+69=? 64. 64+55=? 84. 25+88=? 45. 78+73=? 65. 29+33=? 85. 36+54=? 46. 65+63=? 66. 36+93=? 86. 97+79=? 47. 73+39=? 67. 74+67=? 87. 88+69=? 48. 55+97=? 68. 36+56=? 88. 53+76=? 49. 77+67=? 69. 75+27=? 89. 95+84=? 60. 59+63=? 70. 33+13=? 90. 67+96=? 61. 78_|_64=? 71. 29+19=? 91. 44+39=? 62. 59+33=? 72. 98+29=? 92. 29+88=? 63. 98+97=? 73. 29+35=? 93. 74+86=? 54. 65+36=? 74. 18+62=? 94. 49+57=? 65.21+17=? 75.79+84=? 95.64+88=? 66. 76+93=? 76. 56+33=? 96. 98+77=? 67. 54+64=? 77. 49+54=? 97. 55+76=? 68. 79+98=? 78. 59+39=? 98. 74+84=? 69. 63+64=? 79. 28+97=? 99. 85+99--^? Write the multiplication table as follows: 1. 2 times 1 or once 2 is 2. 2 times 2 are 4. 2 times 3 or 3 times 2 are 6. 2 times 4 or 4 times 2 are 8. Continue this to 12. 2. Write the 3 times table to 12 in the same way. 3. Write the 4 times table to 12 in the same way. 4. Write the 5 times table to 12 in the same way. 6. Write the 6 times table to 12 in the same way. 6. Write the 7 times table to 12 in the same way. 7. Write the 8 times table to 12 in the same way. 8. Write the 9 times table to 12 in the same way. 9. Write the 10 times table to 12 in the same way. 10. Write the 11 times table to 12 in the same way. 11. Write the 12 times table to 12 in the same way. MULTIPLICATION. IV. MULTIPLICATION 22. Multiplication is a short method of adding, when the same number has to be repeated any number of times. X is the sign. 3X^18, reads, three times six equals eighteen. 1. To find the sum of 123+123+123, by additions we would enter the three amounts as before, and add for the result. In multiplication, we multiply each figure of the num- 123 ber to be increased by the number which indicates 3 liow often the repetition is to be made, thus: . 3 times 3 are 9; put 9 in the unit's place. 369 3 times 2 are 6; put 6 in the ten's place. 3 times 1 are 3, which put in the hundred's place. The result is 869, as it would have been by addition. TERMS. 23. The number to be multiplied is called the multipli- cand, the number by which it is multiplied, the multiplier, and the number produced by multiplying, the product. The multiplicand and multiplier are also called factors. T, (123 Multiplicand. FACTORS | jj 369 Product. 2. To find the product of 1496 by 7. Here we say 7 times 6 are 42; write 2 under the 7. 1496 Then 7 times 9 are 63, and the 4 we carried make 67; 7 write 7 and carry 6. 7 times 4 are 28 and G are 34; - write 4 and carry 3. 7 times 1 are 7 and 3 are 10. 10472 Ans. 10472 50 NELSON'S COMMON-SCHOOL ARITHMETIC. 3. 2146X2= 4292 4. 3178X3= 9534 5. 4167X4=16668 9. 5189X5=? 10. 7864X6=? 11. 2875X7=? Sum, 93254. 6. 21007X5=? 7. 31497X<5=? 8. 17843X7=? Sum, 418918. 12. 41679X 8=? 13. 98765 X 9=? 14. 73149X12=? Sum, 2100105. Observe to point off the cents in the products of the following : 15. $21.37X7=? 16. $117.49=8=? 17. $317.00x0? Amount, $3942.51. 21. $678.39X11=-? 22. $467.28X12=? 23. $999.99X 9=? Amount, $22069.56. 27. $47.531 X 9=? 28. $716.145X11=?' 29. $9871.321x12=? Amount, $126761.226. 33. $9057.179X12=? 34. $7898.796X 9=? 35. $5970.463X11=? Amount, $245450.405. 39. $7161.213x11? 40. $1409.796X12=? 41. $9393.678 X 9=? 42. $4131.196X10=? Amount. $221545.957. 18. $10.73X9? 19. $117.07X6=? 20. $307.49x7=? Amount, $2951.42. 24. $671.49X10=? 25. $857.37X11? 26. $1096.49X12=? Amount, $29303.85. 30. $1670.053x7=? 31. $2199.989X9=? 32. $7186.739x8=? Amount, 88984.184. 36. $793.179X9=? 37. $987.970X8=? 38. $213.219X7=? Amount, $165349.04. 43. $3714.291 X^? 44. $7965.379X8=? 45. $3768.219x7=? 46. $ 419.367X6=? Amount, $126045.386. MULTIPLICATION. 51 47. 2785X357. We have here three multipliers seven, fifty and three hundred. 2785X7= 19495 19495 2785X5 tens= 13925 tens, or 139250 ' 2785x3 hundreds= 8355 hundreds, or 835500 Total products, 994245 This operation is contracted by arranging the figures 2785 as in the margin, and writing the first figure of the 357 products of the units in the unit's place and the others to the left, of it; the first figure of the product of the tens in the ten's place, or under its own multiplier, 5; oore and the first figure of the product of the hundreds in the hundred's place. 994245 Either factor may be used as a multiplier in the fol- lowing exercises? 48. 3170X 178=? 51. 2896x6789=? 49. 6184X1794=? 52. 7109x9998=? 50. 3867X3784=? 53. 2345x3979=? Total, 26291084. Total, 100067481. To THE TEACHER. Blackboard exercises in concert may be given in the following manner: Writing a line of figures, 379463875426, the teacher would lead by multiplying the second by the first, the third by the second, the fourth by the third, etc., without speaking the process. Pointing to 7, he would say 21; to 9, 63; to 4, 36, etc. To instruct in "carrying," the same line may be used by point- ing to the third figure and performing the following operation mentally: 3X"-f 9=30. Pointing to 4, he would say 67; to 6, 42; to 3, 27. To produce rapidity of thought and action, exercises of this kind ought to be frequent, and the teacher should lead, taking care that the whole class follows. Such exercises as these may be profitably continued throughout the entire course of study. 52 NELSON'S COMMON-SCHOOL ARITHMETIC. 54. 6789X2164=? 55. 1578X 753=? 56. 9409X6781=? 57. 2783X4679=? Total, 92703716. 62. 420001000 109608 58. 8976X7659=? 59. 3968X6483? 60. 7689X2197=? 61. 6784X7898=?. Total, 1649444493. 109608 420001000 3360008000 2520006 3780009 420001 109608 219216 438432 Product, 46035469608000 Product, 46035469608000 The multiplier of the ten's place in the first operation being 0, we passed it, and multiplied by the 6 hundreds. In the second operation we passed the ten's, hundred's and thousand's places for the same reason.* 63. 12346X30010=? . 64. 7684X10900=? 65. 6787 X 3009=? Total, 474681143. 69. 2000X 7010=? 70. 3160X10096=? 71. 2178X90909=? Total, 243923162. 66. 4967X 6007=? 67.. 5896X900707=? 68. 7649X 66080=? Total, 5845851161. 72. 1009x90910=? 73. 21678X21006=? 74. 31784X 7009=? Total, 769870314. 24. To multiply ~by 10, 100, 1000, etc., we have simply to annex as many ciphers to the multiplicand as there are in the multiplier. *If the learner will simply observe to write the first figure of each product under its own multiplier, he will have no difficulty in mul- tiplying where there are ciphers. For instance, the first figure of the product by 2, in the second example, is immediately under the 2. MULTIPLICATION. 53 35X10=350. Proof 35 10 Ten times 5 are 50, and 10 times 3 are 30 and 5 are 35, making 350. 350 75. 1G5X 10= 1650 78. 413X 10=? 76. 165X 100= 16500 79. 1716X 100=? 77. 165x1000=165000 80. 9417x1000=? Total, 9592730. 81. 374X 100=? 84. 9361 X 10=? 82. 268X 1000=? 85. 7342X 100=? 83. 189X10000=? 86. 8654x1000=? Total, 2195400. Total, 9481810. 25. To multiply dollars, cents and mills, we remove the decimal point to the right. 87. What is the product of $279.373 by 10? Am. $2793.73. EXPLANATION. By multiplying the mills by 10 we make them cents, by multiplying the cents by 10 we make them dimes, and by multiplying the dimes by 10 we make them dollars. 88. $145.373 XlOO=$14537.3, or 14537 dollars and 3 dimes or 30 cents. 89. $356.14,5 X 10=? 92. $317.98,7x 100=? 90. $178.91,3X100=? 93. $679.97,6X 10=? 91. $463.97,8X100=? 94. $7193.44,5x1000=? Amount, $67850.55. Amount, $7232043.46. 95. $713.71,4X 100=? 99. $131.71,2x1000=? 96. $165.79,3X1000=? 100. $724.26,8 X 100=? 97. $786.47,5X 10=? 101. $413.16,4X 10=? 98. $130.14 X 100=? 102. $236.21 X 100=? Amount, $258043.15. Amount, $231891.44. To THE TEACHER. Mental exercises on this subject should suc- ceed these written ones. NELSON'S COMMON-SCHOOL ARITHMETIC. HOME EXERCISES. I. Commencing at 13, write the 2 times table to 19 in this way : 2 times 13 are 26; 13 times 2 are 26. 2 times 14 are 28 ; 14 times 2 are 28. 2. Write the 3 times table in the same way. 3. Write the 4 times table in the same way. 4. Write the 5 times table in the same way. 5. Write the 6 times table in the same way. 6. Write the 7 times table in the same way. 7. Write the 8 times table in the same way. 8. Write the 9 times table in the same way. 9. Write the 13 times table to 9 as follows: 13 times 2 are 26; 2 times 13 are 26. 13 times 3 are 39; 3 times 13 are 39. 10. Write the 14 times table in the same way. II. Write the 15 times table in the same way. 12. Write the 16 times table in the same way. 13. Write the 17 times table in the same way. 14. Write the 18 times table in the same way. 15. Write the 19 times table in the same way. 1. Write the 2 times table to 19 as follows: 2 times 3 are 6; 2 times 13 are 26. 2 times 4 are 8; 2 times 14 are 28. 2 times 5 are 10; 2 times 15 are 30. 2. Write the 3 times table in the same way. 3. Write the 4 times table in the same way. 4. Write the 5 times table in the same way. 5. Write the 6 times table in the same way. 6. Write the 7 times table in the same way. 7. Write the 8 times table in the same way. 8. Write the 9 times table in the same way. MULTIPLICATION. 55 EXERCISES IN MULTIPLYING THE " TEENS."* 100. 2357X13 and 14? 104. 7890x10 and 18? 101. 5398X15 and 16? 105. 2164x17 and 16? 102. 6532X17 and 18? 10(1. 3165x16 and 15? 103. 7654X17 and 19? 107. 2137x15 and 14? Total products, 735141. Total products, 523430. 108. $45.67X16 and 15? 112. $76.54 x!6 and 17? 109. 14.59X15 and 14? 113. 57.352x19 and 15? 110. 23.08X13 and 14? 114. 67.185x18 and 17? 111. 21.87X17 and 19? 115. 45.375x16 and 18? Total products, 3249.36. Total products, 8370.013. 116. 137.67 Xl7andl8? 120. 43.165X10 and 16? 117. 216.031X15 and 18? 121. 933 Xl8andl5? 118. 131.75 Xl6andl6? 122. 61.751x16 and 17? 119. 231.35 Xl7andl9? 123. 311.155x14 and 13? Total prod., 84595.96. Total prod., 42738.743. f PRINCIPLES OF MULTIPLICATION. 26. When two numbers are to be multiplied together, we use for the multiplier that which will produce least figures in the operation. This will be accomplished by selecting the smaller number, except where there are many ciphers, as in Ex. 62. 27. If a number of articles and the price of one article be multiplied together, the product will be the price of all at the same rate. 3 yards of muslin at 20 cents. 20X'3=60 cents. If the price of one be in cents, the price of all wilf be in cents. If in dollars, the price of all will be in dollars. * Every boy designed for business pursuits ought to commit to memory the multiplication table up to 19 times, inclusive. 56 NELSON'S COMMON-SCHOOL ARITHMETIC. 28. The number of articles contained in any box, bale, package, etc., multiplied by the number of boxes, bales, etc., each containing a like number, will give the number of articles in all. In a box there are 30 articles; how many in 20 such boxes? 30X20=600 articles. 29. A number multiplied by itself is squared, or raised to the second power, and the second power multiplied by the same number is cubed, or raised to the third power. The sign is a small figure on the right of the number, thus, 5*, which indicates that 5 is to be raised to the fourth power, and is equal to 5x5X^X5, or 625. 30. Feet multiplied by feet and yards multiplied by yards produce square feet and square yards. 12 feetXl2 feet=144 square feet. 31. Any number of feet multiplied by the number of inches in one foot will give the number of inches in all the feet. Pounds multiplied by the number of ounces in one pound will give the number of ounces in all the pounds, and so with numbers of any other denomination. How many inches in 37 feet? 37X12=444 inches. 1. What is the price of 37 bushels of corn at 37 cents per bushel? 2. What should I pay for 357- yards of broadcloth at $2.?5 per yard? 3. Find the cost of 325 acres of land at $57 per acre. 4. In 320 bales of cotton there are 460 Ibs. each ; how many in all? 5. In 557 pieces of muslin there are 35 yards each; how many in all? MULTIPLICATION. 57 6. A ship laden with flour has 7950 barrels on board, and in each barrel there are 196 Ibs.; how many pounds in all? 7. In a bushel of dried apples there are 25 pounds; how many are there in 37 bushels? 8. A barrel of flour weighs 196 pounds; what is the weight of 325 barrels? 9. What will be the weight of 134 bushels of wheat when 60 Ibs. are allowed to the bushel? 10. Find the cost of 379 boxes of cheese, each of which weighs 22 pounds, at 25 cents a pound. 11. A box of buttons contains a gross; how many but- tons are there in 59 boxes? 12. A merchant sold 135 barrels of flour at $6.75 apiece, and with part of the money bought 369 bushels of coal at 25 cents; how much money had he left? 13. In 236 yards how many inches? 14. How many quarts are there in 27 bushels? 15. At 23 cents a quart, how much money can be real- ized on 18 bushels of strawberries, allowing one quart for loss in measuring? 16. A huckster bought two barrels of apples, each con- taining 3 bushels, at $6 a bushel, and sold them at 87 cents a half peck ; allowing half a peck for loss in meas- uring, did he gain or lose? and how much? 17. There are 12 inches in a foot and 3 feet in a yard; how many inches are there in 357 yards? 18. How many quarts of wine are there in 6 barrels, each of which contains 42 galls.? how many in 35 bar- rels? how many in 163? Answers without their denominations: 13225, 864, 819, 8496, 208450, 8496, 8040, 925, 98175, 21397, 18525, 147200, 63700, 19495, 1558200, 489, 1369, 12852, 34272. 58 NELSON'S COMMON-SCHOOL ARITHMETIC. 19. Find the cost of 117 bushels of apples at 35 cents a peck. 20. Find the cost of 237 bushels of potatoes at 42 cents a peck. 21. Bought 46 horses at 125 apiece, and sold them for 85900; did I lose or gain, and how much? 22. At 5 cents apiece, what will 22 gross of eggs cost? 23. In 15 acres how many square yards? 24. Find the cost of 35 barrels of molasses at 33 cents a gallon, each barrel containing, on an average, 41 galls. 25. In 257 cords of wood how many solid feet? 26. How many bushels of coal in 17 wagons, each car- rying 50? and what will be the cost at $17 a load? 27. How many yards are in a box of silk containing 35 pieces, each piece measuring 52 yards? 28. In a case of muslin there are 45 pieces, each 31 yards; what will be the cost of it at 55 cents a yard? 29. A bushel of hemp-seed weighs 44 pounds; what will be the weight of 137 bushels? 30. How many leaves are in 67 reams of paper? 31. How many half-pence are in 527 pounds sterling? Answers promiscuously arranged: 72600, 16380, 5175, 150, 15840, 39816, 47355, 64320, 76725, 14450, 1820, 32896, 6028, 252960. HOME EXERCISES. 1. 3+ 4 2=1 2. 5 3+ 7=? 3. 9>< 4_ 3 = ? 4. 7>< 6+ 8=? 6. 6+ 9X 6r=? 6. 9 8 1=1 7. 7+ 3X 6=? 8. 9-f- 8+ 3=? 9. 9>< 7_ 4^? 10. 5+ 5 3=? 11. 8X 4+ 2=1 12. 9 3 2=1 13. 16+12 5=,? 14. 23-}-20 1=? 15. 1916+ 1=1 16. 2712-1- 8r=? 17. 29+ 2 7=1 18. 40+54 3? MULTIPLICATION. 19. 35 15X 6=? 20. 64 -fSO 2=? 21. 971?,+ 7=? 22. 55+43 8=? 23. 63+65+ 2=1 24. 19+ 6 5=? 25. 29+ 6 5=? 26. 35+25+15=? 27. 18+2210=? 28. 54+16+10=? 29. 93+1720=? 30. 44+16+32 = ? 31. 12+18+33=? 32. 27+33+54=? 33. 44+1615=? 84. 1912+28=? 36. 55+44+21=? 36. 23+60+47=? 37. 64+15+36=? 38. 24+46+33=? 39. 56+44+71=? 40. 19+33+21=? 41. 77+44+23=? 42. 45+31+55=? 43. 37+13+34=? 44. 66+1412=? 45. 44+76+33=? 46. 92+1810=? 47. 73+10+67=? 48. 21+39+22=? 49. 17+63+14=? 50. 29+33+19=? 51. 3X15+15=? 52. 7+16X 3=? 53. 57 17X "^? 54. 19+11X 6=? 55. 45+15X 9=? 56. 84+26X12=? 57. 21+18+15=? 58. 45+1611=? 59. 73+1316=? 60. 55+45+35=? 61. 79+60+30=? 62. 24+71+ 5=? 63. 39+1110=? 64. 9919+12=? 65. 77+7914=? 66. 85+9611=? 67. 77+6320=? 68. 45+6522=? 69. 33+7320=? 70. 97+93+50=? 71. 79+91+93=? 72. 56+64+49=? 73. 73+65+37=? 74. 93+65+35=? 75. 97+63+47=? 76. 45X 3+45=? 77. 35X 4+30=? 78. 29X 614=? 79. 99X 327=? 80. 84 14X 7=? 81. 75X 3 6=? 82. 36X 414=? 83. 56X 3+50=? 84. 75 16X 6=? 85. 35X1220=? 86. 42X10+22=? 87. 57X1214=? 88. 4515X30=? 89. 64X 520=? 90. 18X 628=? 91. 36X12+72=? 92. 42X12+12=? 93. 35X20+12=? 94. 64X3214=? 95. 33X2112=? 96. 14X1213=? 97. 12+33 6=? 98. 75X16 4=? 99. 39X12+ 4=? 100. 5616+ 6=? 101. 35X15X 6=? 102. 3522X12=? 103. 64+2210=? 104. 75X9737=? 105. 64X14 4=? 106. 3616X96=? 107. 5523X20=? 108. 4919X84=? 109. 64X4515=? 110. 976717=? 111. 37+16 6=? 1. Write the 2 times table to 9 as follows: 2 times 2 are 4 ; 2 is contained in 4, 2 times. 2 times 3 are 6; 2 is contained in 6, 3 times. 2 times 4 are 8; 2 is contained in 8, 4 times. 2. Write 3 times table to 9 as above. 3. Write 4 times table to 9 in the same way. 4. Write 5 times table to 9 in the same way. 6. Write 6 times table to 9 in tlie same way. 60 NELSON'S COMMON-SCHOOL APITHMETiC. 6. Write 7 times table to 9 in the same way. 7. Write 8 times table to 9 in the same way. 8. Write 9 times table to 9 in the same way. 1. Write the division table, as below, to 2 in 19: 2 in 2, 1 time. 2 in 3, 1 time and 1 left. 2 in 4, 2 times. 2 in 5, 2 times and 1 left. 2. Find how often 3 is contained in numbers from 3 to 29. 3. Find how often 4 is contained in numbers from 4 to 39. 4. Find how often 5 is contained in numbers from 5 to 49. 5. Find how often 6 is contained in numbers from 6 to 59. 6. Find how often 7 is contained in numbers from 7 to 69. 7. Find how often 8 is contained in numbers from 8 to 79. 8. Find how often 9 is contained in numbers from 9 to 89. V. DIVISION. 32. Division is the method of calculation used to sep- arate numbers into equal parts. 33. Division may be short or long. It is short when the process of finding the product and remainder is per- formed mentally, and long when the process is written. 34. The sign is -f-, which placed between two numbers indicates that the one on the left is to be divided by the one on the right. 6-^3, reads, six divided by three. Division is also indicated by a curved line between the numbers, thus: 3)6; and by a straight line, with one number above and the other below, as |, called the frac- tional form. The period is also used to indicate division. .5 shows that 5 is divided by 10. 35. TERMS. The number divided or to be divided is the dividend. DIVISION. 61 The number by which the division is performed or to be performed is the divisor. The number which shows how many times the divisor is contained in the dividend is the quotient. The number left after dividing, the remainder. Divisor, 3)16784 Dividend. Quotient, 5594 2 Remainder. SHORT DIVISION. 1. To divide 738 by 3. 3)738 EXPLANATION. 1. Commencing at the left, we find - how often 3 is contained in 7 hundred, which is 2 hun- 246 dred times, with a remainder of 1 hundred. The 2 we write in the hundreds' place. 2. This 1 hundred, with the 38, gives a remainder of 138 to be divided. To divide this, we consider the 13 as tens. In 13 tens 3 is contained 4 times, and 1 ten left as a remainder ; so we write the 4 in the tens' place. 3. This 1 ten, with the remaining 8, gives 18 to be divided. In 18, 3 is contained 6 times, which, being written under the 8, gives the result, 246. Until he becomes familiar with the process, the learner may write the remainders in small figures, as in the fol- lowing example. 2. 8)1 3 5 2 7 3 6 4 8 3 3 7 5 6 1 *7 163460470 21 EXPLANATION. 1. Commencing at the left we find how many times 8 is contained in 13. The answer being 1 time, with a re- mainder of 5, we write the 1 under the 13 and the 5 before the 0. 2. This 5 taken with the makes 50, in which 8 is contained 6 times, with a remainder of 2. 3. This 2, with the 7, makes 27, in which 8 is contained 3 times, with a remainder of 3. 4. When 3 in the dividend is reached, it is found that 8 is not 62 NELSON'S COMMON-SCHOOL ARITHMETIC. contained in it, so a cipher is placed under it and 3 considered a remainder.* Divide the following: 3. 134615379-4-2 4. 21637298452-5-3 5. 59368217755--4 6. 1416823687949-f-5 Rem 7. 13645217--6 5 8. 23176841-:-7 2 9. 47896739--8 3 10. 89765432-J-9 8 Quotients. Rem. Am. 673076891 Am. 72124328171 Ans. 148420544383 Am. 2833647375894 Rem. 11. 36174573110 1 12. 21376495211 5 13. 17896152112 1 14. 34567890012 Total quotients, 21546207 and 99327785. 15. 80923751763=? 19. 33376231825 7=? 16. 7316305416- 17. 351923649755=? 18. 13138539444 23. 6542050675611=? 24. 9645148784412=? 25. 31463215726 7=? 26. 21347096543 8=? Total, 2114807495312. -4=? 20. 72143250072 8=? 21. 30608807751 9=? -6=? 22. 8172695439010=? Total quotients, 13754764315 and 25359613454. 27. 57S312908-f-3=? 28. 483796459-^-4=? 29. 761147355-^-5=? 30. 123450678-4-6= ? Total, 4865246673. 31 to 34. Divide 874109630175 by 2, 3, 4 and 5. Sum, 11217740253904. * PROOF OF DIVISION. Division may be proved by multiplying the quotient by the divisor and adding the remainder. Take the last example: 1634604702-1 12076837617 DIVISION 63 35 to 38. Divide the same number by 6, 7, 8 and 9. Sum, 47694473868421. 39 to 41. Divide the same number by 10, 11 and 12. Sum, 2397179440329. 42. 710084973-r- 7=? 46. 31463574213--7=? 43. 394789006-f- 8=? 47. 91678543210-=-8=? 44. 361007839-5- 9=? 48. 76303074368--9=? 45. 909738697--10=? 49. 21356703648-f-6=? Sum, 28187518617. Sum, 279921841995. 36. The quotient of a number divided by 2 is the (one-half) of it ; divided by 3 it is the -| (one-third) ; by 4, the \ (one-fourth); hence, to find the , \ or \ of a number, we have simply to divide by 2, 3 or 4.* Let it be requited to find the cost of 2| yards of cloth at $3 a yard. $3 =cost of 1 yard. 2 6 cost of 2 yards. \^=cost of \ yard. $7=cost of 2^ yards. Here, to multiply by 2 j, the 3 had to be divided by 2. 50. -J of 3716=1238g f 54 - 7 of 34161143764= - 61. -I of 1367= - 1 55. of 37897181237= - | 62. ^ of 7854= - 1 56. J of 16872352168= - | 53. J of 879= - 1 57. T V of 34564185432= - / 3 *The learner will be particular to observe that finding the J or J of a number is not dividing by one-half or one-third, but simply finding one part of something divided into two or three parts which is multiplying by one-half or one-third. "j The remainder in this example being 2, we write it as f, which indicates that 2 is divided by 3, or that it is 2 parts of something divided into 3 parts. The remainders only of the questions which follow will be given. 64 NELSON'S COMMON-SCHOOL ARITHMETIC. 58. $14567.85X9| is how much? 1456785 cents. 13111065 485595 13596660 or $135966.60 The decimal point was removed before dividing and replaced after the operation was performed. 59. J- of $21372 =? 63. of $67849132.87=? 60. of $13744 =? 64. \ of $16493178.00=? 61. i of $73176.35=? 65. \ of $23610934.10=? 62. \ of $14537.07=? 66. T V of $12310985.47=? Total, $25092.50. Total, $14579927.51. 67. $345.78 X 37 ^=how much? 34578 37A 17289 = \ of multiplicand. 242046 = 7 times do. 103734 =30 times do. 1296675 =37-| or $12966.75. 68 and 69. $146.82x8-*=? $1713.14x6{ = ? Sum, $11930.62,1. 70, 71 and 72. $4563.28X451, 16J and 18. Sum of products, $365062.39,91+ 1. 73, 74 and 75. $21763X14}, 15, 29|. Total, $1274430.916 c j. 76, 77 and 78. $7649.14X76J, 96J, 86-j. Total, $1977090.20j + . 79 to 81. $3146X2^, 6^, 12^. Total, $67639.00. 82 to 84. $15G7X^, 5," llU. $39175.00. 85 to 87. $7864X6|, 71, 37^. $400670.80. DIVISION. 65 88 to 90. $71684.25x8|, 7|, 5'. 91 to 93. $89647.86 X 4, 2, s|. 94 to 96. $79943.52 X , 6, 71. Amount, $3327494.19. 97 to 100. If a steamboat is worth $3456, what will | be worth? What f ? What f ? What f ? Total, $6912. REMARK. 2 fifths will cost 2 times as much as 1 fifth; 3 fifths, 3 times as much as 1 fifth, etc. 101 to 105. T V of $155367 is how much? -l=? |=? f=? |=? Total, $97720.910f 37. 2^0 divide by 10, we point off one figure on the right, by 100, two figures, by 1000, three figures. Those on the left will be the quotient, those on the right the remainder. 500 divided by 10=50.0. $500 divided by 100=$5.00. REASON. By pointing off one figure we remove all the figures one place further to the right, so that the tens stand where the units were, and are units, the hundreds where the tens were, and are tens. 2. It will be observed that the number pointed off corresponds with the number of ciphers in the divisor. For 10 we point off one figure; for 100, two; for 1000, three. 31456-^-100=314.56, or 314^ 3. Observe, also, that a decimal fraction, as .65, is changed to a common fraction by removing the point and writing the figure 1, with as many ciphers annexed as there are figures in the decimal: 106. 134567 10=? 107. 34578 100=? 108. 713641000=? 109. 21789 100=? 110. 2163 100=? 111. 78651000=? 112. 113. 114. 115. 116. 9876 10=? 3453100=? 6543 10=? 2197100=? 1763 10=? 117. 7974100=? Total quotients, 14121.229 and 1954.44. 38. To divide dollars and cents, the decimal point is re- moved to the left, which is the same as pointing off. 6 66 NELSON'S COMMON-SCHOOL ARITHMETIC. To divide by 10, it is removed one figure; by 100, two figures. $55.10 divided by 10=$5.510. $167.56 divided by 100=81.67,56.* Divide the following: 118. $457.87-4- 10. 122. $473.041000 and 100. 119. $1677.45-4- 100. 123. $15.17 10 and 100. 120. $6109.88^-1000. 124. $16.57 100 and 10. 121. $14999.99-4- 100. 125. $106.07-4- 100 and 1000. Total, $218.66. Total, $9.85,9 126. Divide the following sums of money by 100: $645, $1678.25, $87493.57, $16453.27, $1998.38, $643.24, $2168, $4137.54. Total answer, $1152.16,9. 39. It often happens that there are not as many figures to cut off as there are ciphers in the divisor. In such cases we prefix ciphers to the dividend to make up the number. Divide $5. by 100. Ans. .05. EXPLANATION. This is the same as removing the decimal point two places to the left, as above. The $5 had the decimal point on the right of the 5; it is now two places further to the left, and therefore is divided by 100. The cipher, in this case, as elsewhere, possesses no value. 127. $5-4- 10=.5 132. $0.03-- 10=? 128. $3^- 100=.03 133. $0.02-4-100=? 129. $4-f-1000=.004 134. $0.14-=-100=? 130. $50-4-1000=.05 135. $3.16-f-100=? 131. $457-4-1000^.457 136. $21.30-4- 10=? Total, 1.041. Total, 2.1662. *NOTE. The value of each and all of the figures decreases ten- fold for every figure the decimal point is removed to the left. The $5 in the first example became 50 cents and the 10 cents be- came 10 mills or 1 cent, making the answer 5 dollars, 51 cents, not 6 dollars, 510 cents. The second answer is 7 dollars, 67 cents, 5 mills and / of a mill, or $1.07,5^. ^~. \ DIVISION. 67 137. Divide the following sums by 100: 3 cents, 33 cents, $3.33, $33.33, $333.33, $3333.33. Total, $37.03,68. 40. To divide by 20, 300, 5000, etc., we point off as many figures in the dividend as there are ciphers in the divisor, and divide by the 2, 3, 5, etc. The figures pointed off will form part of the remainder.* 138. Divide 317745 by 500. 5 1 00)3177 1 45 EXPLANATION Pointing off two figures, we 635245 divide by 100; what is left we divide by 5. 635 ?~ 139. 467831-4- 20=23391^1 142. 716849-=-700=? 140. 71 6893^-300=2389^9 3 143. 897653--900=? 141. 417368--500= 834|g 144. 49673--- 80=? Total quotients, 2641. Total rein., 1573. The answers to the following are required in dollars, cents and mills, omitting the remainders: $2131.51500=? Reduced to mills, it is 2141510. 5 | 00)21315 | 10 4263 ^=$4.26,3^. 145. $13764.75-4-50=? 149. $16789.37-- 80=? 146. $73968.23-4-60=? 150. $67859.67-^-900=? 147. $37437.18^-90=? 151. $54168.23^-700=? 148. $18964.20-4-80=? ' 152. $78910.00-4-600=? Total, $2161.11,8. Total, $494.16,5. 153 to 157. Divide the following sums by 20, and give the answers as above: $1367.25, $3143.57, $2345.87, $34.57, $45679.44. Total, 2628.53,3. *When dividing dollars and cents, reduce them by erasing the decimal point and annexing ciphers if necessary. $357/27 ~500~ ? Reduced to cents the dividend is 35727. 5 | 00)357 | 27 ~Kffi 68 NELSON'S COMMON-SCHOOL ARITHMETIC. 158 to 165. Divide $34567.25 by 10, 12, 20, 100, 30, 50, 70 arid 90. Total, 11132.84,6. 166 to 176. Divide $367897.87 by 100, and the quotient by 10, 20, 30, 40, 50, 60, 70, 80, 90. Total, $4719.74,4. 177 to 187. Divide $17654.37 by 100, and the quotient by 3, 10, 7, 40, 30, 50, 70, 90 and 80. Total, $298.78. 188 to 196. Divide $314937 by 100, and multiply the quotient by 7; then divide the quotient by 30, 60, 40, 12, 9, 80, 90. Total, $22968.52,7. VI. PERCENTAGE.* 41. Percentage is the method of reckoning by Jiun- dredths. 1 per cent, is the one hundredth part, 2 per cent, twice that amount, 3 per cent, three times that amount. 42- The sign is %. 25% signifies 25 per cent. 43. To compute percentage, or, in other words, to find any rate per cent., we first find one per cent., and multi- ply it by the given rate. 1. To find 5% of 350. One per cent, is the number divided by 100 or 3.50. 5 per cent is 3.50x5=17.50 or 17^% or l7|i 2. Find 6% of 3572. OPERATION. 35.72=1 per cent. 6 214.32 or 2U^=6%. 3. 6% of 3146 is how much? Ans. 188.76 or 4. 5% of 1937 is how much? Ans. *This rule is of such general utility, is so simple in its appli- cation, and so strictly belongs to the subject of division, that I can not refrain from introducing it in this place. PERCENTAGE. 5. 7% of 3176 is how much? 6. 9% of 7854 is how much? 7. 8% of 396 is how much? 8. 4% of 243 is how much? Amount, 970^. 13. 20% of 3161=? 14. 33% of 798=? 15. 55% of 654=? 16. 19% of 321=? Amount, 1316^. 21. 12J% of 167=? 22. 151% of 364=? 23. 371% of 910=? 24. 50J% of 693=? Amount, 7< 9. 4% of 1300=? 10. 6% of 367=? 11. 9% of 463=? 12. 8% of 6735=? Amount, 654.49. 17. 30 % of 4541=? 18. 23 % of 147=? 19. 60 % of 7163=? 20. 33% of 4371=? Amount, 7150^. 25. 2% of $320=? 26. 3% of $976=? 27. 5% of $8900=? 28. 7% of $6540=? Amount, $938.48. 33. 20 % of $1361=? 34. 451% of $316=? 35. 17 % of $2163=? 36. 19J% of $1723=? Amount, 1119.67,5. 37. 25% of $264.50 is how much? 2.6450=1% J 25 132250 52900 29. 7% of $327=? 30. 9% of $100=? 31. 12% of $978=? 32. 25% of $179=? Amount, $194.00. 66.1250 or $66.12,5=25%. 38. 3% 39. 5% 40. 6% 41. 7% of $674.75=? 43. of $198.45=? 44. of $786.70=? 45. of $14.13=? 46. 42. 12% of $1.19=? 47. Amount, $78.49,8 T V 6% 12% 11% 40% 50% of $397.25=? of $187.17=? of $710.00=? of $1678.00=? of $7764.82=? Amount, 4678.00,5 T 4 n . 70 NELSON'S COMMON-SCHOOL ARITHMETIC. 48. 37J% of $461.75=? 51. \*\% of $610.18=? 49. %\\% of $198.18=? 52. \\% of $114.14=? 50. 62%% of $213.07=? *Amount, 539.60,8 r 6 5 ff . GENERAL EXERCISES. 1. Divide $367.22 equally among 7 persons. Each will have ^; divide by 7. 2. 12 horses, of equal value, cost $3456.24; what was the cost of each? 3. If 5 men accomplish a piece of work in 320 days, how long will it take one man to do it? 4. In a year there are 365 days; how many weeks will that make? 5. A ship sails 84 miles in 12 hours; what is her aver- age speed per hour? 6. How many shillings in 13456 pence? how many pounds? 7. If 30 bricklayers can erect the walls of a house in 120 days, how long will it take 12 to do it? 8. How long will it take a writer to copy a speech of 22340 words, if he writes 40 words a minute ? 9. If $345.72 be divided among 12 persons, how much will each receive? 10. In a pound there are 20 shillings; how many pounds are there in 3456 shillings? 11. How many shillings are there in 21345 pence? how many pounds? 12. How many bushels of wheat in 134563 pounds, reckoning 60 pounds to the bushel? Answers: 1600, $52.46, $288.02, 52}, 2242f, 7, 16, 88 18 9, 9, 172 16, 300, 5581 $28.81, 1121 T 4 2 , 1778 9. *The Teacher should give numerous oral exercises on this rule, else scholars will be apt to err in pointing the products. GENERAL EXERCISES, 71 13. A and B ar-e in partnership, in which A invests $30 and B 20. They make 14; how much should each receive ? A's share is 30 parts. J3 $ share is 20 parts. Together they have 50 parts. 1 part of $14 divided into 50 parts=28 cents; and 20 parts= $5.60, nnd 30 partsr=r$8.40, which, added together $14. 14. 2 men trading horses put in each $1200 and $800, and gained $1250; what was each man's share? 15. If 20 men do a piece of work in 30 days, how long will it take 1 man to do it? how long 15? 16. In 3683 oranges how many dozen? 17. If $3678.21 be divided between 7 persons, how much will each receive? 18. The profits of a speculation in which was invested, by 3 persons, $300, $900 and $800, are $919.18; how much should each receive? 19. The cost of muslin is 30 cents per yard; how many yards can be bought for $397 at that rate? 20. Find the cost of the following articles: 125 Ibs. sugar at 27 cents a pound; 37 Ibs. of butter at 37Jc; 2 hams, each weighing 13 and 14 Ibs., at 21c; 115 Ibs. of cheese at 15J c. 21. How much money will buy 37 \ Ibs. of tea at $2 a pound; 150 Ibs. of fish at 8Jc; 57 Ibs. of sugar at 31Jc; 56J Ibs. of lard at 16 c ; 45 Ibs. of soap at 7J c; 31 Ibs. of candles at 16 c? 22. A lady bought the following goods and paid for them out of $100; how much had she left? 8J yds. of French merino @ $2.25; 2 pieces brown muslin, 33 ard 50J yds., at 35 c; 1 bonnet at $7.50; 1 shawl at- $11.50. - Answers : 750, 500, 600, 40, 306f , $23.73,7,* , $33.11, $525.45f, 1323& $71.12, $367.67,2, $137.87;T $413.63,1. '72 NELSON'S COMMON-SCHOOL ARITHMETIC. VII. LONG DIVISION* THE previous operations in division were performed al- most mentally, the learner writing only the quotients. That method is preferable when the divisor is found in the tables, or can be reduced to a number contained in them, as 500, 1200; otherwise the operation would be too difficult and tedious to perform mentally. 44. Long Division has then to be used, which consists in writing the products and remainders as well as the quotients. The better to illustrate this method, an ex- ample which can be solved by short division is selected. 1. Divide 3147 by 6. Divr. Dii-d. Quot. EXPLANATION. 1. To perform this operation, o)ol47(5^4g we try, as before, how often the divisor is con- tained in part of the dividend. 6 is contained 14 in 31, 5 times, and writing 5 in the quotient, we 12 multiply the divisor by it and write the product (30) under 31. 2. We now subtract the 30 from the 31 as we v/r would perform an operation in subtraction. The remainder is 1. Instead of supposing this 1 to stand before the 4 in the dividend, we bring down the 4 to it, mak- ing 14. 3. 6 in 14 is contained 2 times, which 2 we write in the quotient, multiply it upon the divisor (6) and write the product underneath. 4. Subtracting this 12 from the 14, we have a remainder of 2, to which we annex 7 from the dividend, making 27. 5. 6 in 27, 4 times. Writing the 4 in the quotient, we multiply it upon 6, making 24, which we write underneath. 6. Subtracting, as before, we find a remainder of 3, and as there *This, considered the most difficult rule in arithmetic, may be deferred until the learner has passed Easy Fractions. LONG DIVISION. 73 are no more figures in the dividend to bring down, we consider this the final remainder and express it in fractional form, as in short division, |. To enable the learner to comprehend this part of divis- ion more clearly, another example is introduced. 3. Divide 8317517 by 723. 723)8317517(11504} ~ 5 1 The product of the divisor (723) and 1 723* ! I of the quotient. 2 Tno'r 2 The remain der of 831723, with 7 ! annexed. *^ 3 The product of 723 and the second 4 3645 figure of the quotient. 6 3615 4 The remainder of 1087723, with 6 6 orj-j n annexed. 7 2892 5 The P roduct of 723 X 5 ) tne third fig- ure of the quotient. 8 125 6 The remainder, with two figures (1 and 7) annexed. 723 was not contained in 301, so another figure was annexed. 7 The product of 723X4, the fifth figure of the quotient. 8 The remainder. This is represented in fractional form in the quotient. REMARKS. 1. Instead of using the whole divisor in finding a quotient figure, it will generally do to use only the first one or two figures. In the preceding exercise, the first figure alone (7) was used, and in this way: "7 is contained in 8 how many times?" 2. The products should never exceed the numbers above them. Number 3 should not exceed number 2. If on trial, it is found they do so, then a smaller number should be put in the quotient. 3. For every figure brought down from the dividend, there should be one in the quotient. When the divisor is not contained in the new dividend, a cipher should be placed in the quotient and another figure annexed. 4. The divisor can not be contained more than 9 times in the smaller dividends, as 1087, 3645. * The learner should put a mark under each figure brought down to prevent its being taken twice. T 74 NELSON'S COMMON-SCHOOL ARITHMETIC. 4. 71036- 5. 31978- 6. 167864- 7. 9765837- 8. 1763- - 21= 3382^4 - 43=: 743f| - 54= 3108ff - 65=150243|f - 76=? 13. 3167- 14. 71438- 15. 67898- 16. 78637- 17. 1000Q- - 129=? - 320=? - 764=? - 892=? -7109=? 9. 7964- - 87=? 18. 7185- -1990=? 0. 89737- - 98=? 19. 67416- - 144=? 1. 77168- - 19=? 20. 3784- - 642=? 2. 3167- -119=? 21. 14098- - 671=? 52. *$730.45-r-126=? 25 . $89289.61295=? 53. $164.87-i-144=? 26 $21008.97444=? 24. $1710.14-^166=? 27. $10000.00180=? Answers: 8 to 12, inclusive, quotients, 5116; remain- ders, 211. 13 to 16, quotients, 423 and 956. 17 to 21, 498 and 19246. 22 to 24, $39.54; remainder, 6. 25 to 27, $405.53; remainder, 629. When there are ciphers in the divisor, they may be pointed off with a corresponding number of figures in the dividend. 28. 67314968-^-163000 is how much? 163 1 000)67314 1 968(412}|f -g 652 211 163 The figures pointed off in the dividend were annexed to the remainder, forming the 484 fraction, jff |gg. 326 158 29. 12986745 7300=? 30. 8109867018000=? 31. 513643 2500=? Quot's, 6489; Kern., 9858, 32. 7613412-- 37100=? 33. 4567800-;- 20900=? 34. 5632710-^-171000=? Quot's, 455; Hem.; 180222. * Express the dollars in cents before dividing. LONG DIVISION. 75 35. Find the sum, difference, product and quotient of 128097 and 8070; of 1736009 and 4761 ; and 4070391 and 71068, omitting the remainders. Totals, 289282688427, 1033998999, 8268611231 and 29993710999. PRINCIPLES OF DIVISION. 45. If we divide the price of a number of things of equal value by the number, we obtain the price of one. 46. The quotient will usually be in the same name with the dividend or number to be divided. If the dividend be dollars, the quotient will be dollars ; if it be rods, the quotient will be rods. 36. If 75 barrels of flour cost $450, what was the price per barrel? 37. If 125 horses cost $25000, what was the cost of each? 38. If $167809 be divided among 7614 persons, how much should each receive? 39. How much tax should each of 16785 persons pay of a levy of $71683? 40. In 303,656,837 Ibs. of cotton, how many bales, sup- posing each bale to weigh 320 Ibs.? 41. If $79640 be divided among 274 persons, how much will each get? 42. The earth moves round the sun at the rate of 66600 miles an hour; at what rate does it move per minute? 43." If 357 yards of broadcloth cost $1035.30, what was the cost per yard? and what would be the cost of 50 yards at the same rate? Answers arranged promiscuously: 1110, 948, $290.65, $6, $22.30,9, $145, $4.27,7, $200, $2.90. 76 NELSON'S COMMON-SCHOOL ARITHMETIC. EXERCISES IN MULTIPLICATION AND DIVISION.* 1. If 23 yds. of inuslin cost $3.45, what will one yard cost? 2. If 117 men can do a piece of work in 48 days, how long will it take 3 times that number to do it? 3. How many men can do a piece of work in 5 days, that took 10 men 25 days? 4. If a case holds 29 pieces of muslin, how many will it take to hold 7250 pieces? 5. If 15 men can do a certain piece of work in 75 days, how long will it take 1 man to do it? 6. If 7 dozen silver spoons cost $35.35, what will 3 dozen cost? Find the cost of one dozen, then the cost of 3. 7. If f of a ship cost $14602, what will the ^, or the whole ship cost? 8. If | of a piece of property cost $6377, what will -J of it cost? 9. In a cord of wood there are 128 feet, how many cords are in a pile measuring 4 feet wide, 8 feet deep and 100 feet long? 10. In an acre there are 4840 square yards: how many are there in -J- of an acre ? 11. A field contains, 12 acres, and is 660 yards long; how many yards is it in breadth? 12. A tract of 5 acres is 220 yards long; how much should be cut off the breadth to leave 1 acre? 13. -J- of a dozen books cost $7.50; what was the cost per dozen? 14. of a dozen cost 22 cents; what was the cost per dozen? *For answers, see next page. PROPERTIES OF NUMBERS. 77 15. 1^ dozen cost $1.20; what was the cost per dozen? 16. T 7~ of a hundred cost $28; what was the cost per hundred? 17. 2 3 jy of a hundred cost |15.75; what was the cost per hundred ? 18. 1 1 of a hundred cost $3.15 ; what was the price per hundred? Answers: 51107, 3644, 25, 16, 50, 250, 22, 15.15, 605, 60, 88, 80, 33, 100, 4.20, 105, 1125, 15, 88. 47. METHOD OF PROOF. Division and Multiplication beiDg converse operations, the one is proved by the other. DIVISION. PROOF. 38)3715(97 97=quotient. 342 3S=divisor. ~295 776 266 291 29 rem. 3686+tfie rem., (29)=37lb=dividend. MULTIPLICATION. PROOF. multiplier, product, multiplicand. 465 25)11625(465 _25 100 2325 162 930 150 11625 125 125 VIII. PROPERTIES OF NUMBERS. 48. An Integer is any number considered as a whole, as 3, 7. 58, 129. 49. A Fraction is a part of any thing or number of things considered as a whole; ^, -?-, -f-fc. 50. Numbers are divided into Odd and Even. 78 NELSON'S COMMON-SCHOOL ARITHMETIC. An Odd number can not be divided into two equal parts without a remainder, as 1, 5, 57. An Even number can be divided into two equal parts without a remainder, as 4, 10, 68. 51. Numbers are either Prime or Composite, Abstract or Concrete. 52. A Prime number is an original number, or one which can not be produced by multiplying two other numbers together, as 1, 7, 31. 53. A Composite number is one which may be com- posed of two other numbers multiplied together, as 8, which is composed of 2 and 4 multiplied together; and 27, which is composed of 9 and 3 multiplied together.* Exercise. Write out 50 prime and 50 composite numbers. 54. An Abstract number is an unapplied number, or one which conveys the idea of number exclusively, as 4, 15, 47. 55. A Concrete number is an applied number, or one which conveys the idea of something else besides number. The above numbers become concrete when applied as fol- lows : 4 mills, 15 dollars, 47 pounds, and the names, mills, dollars and pounds, are called denominations. 56. A Multiple is a number which contains another number a certain number of times without a remainder. 12 is a multiple of 3 as well as of 2, 4 and 6. 57. A Common Multiple is one which contains two or more numbers a certain number of times without a re- mainder. 12 is a common multiple of 2, 3, and 4. 58. The Least Common Multiple is the least number *NOTE. Since all even numbers are divisible by 2, an even number can not be a prime number, nor can any number ending with 5; it follows, therefore, that every prime number, except 2 arid 5, ends with 1, 3, 7 or 9. PROPERTIES OF NUMBERS. 79 which will contain two or more numbers without a re- mainder. 6 is the least common multiple of 3 and 2, and 18 of 3 and 6, and 24 of 2, 3 and 8. 59. An Aliquot is a number which will divide another without a remainder. The parts of which a multiple is composed are called aliquot parts of that number. 1, 2, 3, 4 and 6 are aliquot parts of 12. 60. Complement: The number required to be added to another to make it equal to a larger. It is usually ap- plied to 100, 1000 or some other power of 10. Taking 87 as a part of 100, the complement is 13, or 50 as a part of 60, the complement is 10. Exercise. Taking 35 as a part of 60, 70 as a part of 100, 18 as a part of 20, 73 as a part of 80, required the complements. 61. Even numbers are divisible by 2 without a re- mainder. 62. If the two right-hand figures of a number are di- visible by 4 without a remainder, the whole nun^ber is also divisible by 4. 63. Numbers ending with 5 or are divisible by 5 without a remainder. 64. If the three right-hand figures of a number are di- visible by 8 without a remainder, the whole number is divisible by 8. 65. If the sum of the figures of any number is divisible by 3 or 9 without a remainder, the whole number will be divisible by 3 or 9. MULTIPLICATION BY ALIQUOTS. 66. To multiply by 2-i, it will shorten the operation if we multiply by 10, which is 4 times too much, and then divide by 4. In the same way we can multiply by any other aliquot of 10, or by aliquots of 100, 1000, etc. 80 NELSON'S COMMON-SCHOOL ARITHMETIC. To multiply 176 by 12|. 8^17600 EXPLANATION: 176 being multiplied by 100 is . - 8 times more than the sum required, so we divide 2200 Ans. by 8> To multiply 379 by 250. 4)379000 379 being mu itiplied by 1000 is 4 times too 94750 Ans. much, so we divide by 4. To multiply $49.75 by 125. 8)4975 000 The $49.75 are considered as cents and mul- />oi o-re tiplied as the preceding. 621 8*5 cents. *! .. . ..,. A/irt-m *r A REMARK. It will lessen the work still more or $6218.75 AM. . . , * 14 . .. , to simply assume the number to be multiplied by 10, 100 or 1000. ALIQUOT PARTS OF 10, 100, 1000. To be committed to memory. ALIQUOTS OF 10. ALIQUOTS OF 100. ALIQUOTS OF 1000. 5 =4 50 =4 142=4 333=i 3i^i 331=4 12= 250=1 2i=3 25 =i 10 = T ^ 166^ I 2 =1 20 =i 8^=^ 125 =^ The pupil can prove the accuracy of his calculations by multiplying in the ordinary way. 4. 140X 12i=? 13. 949 x333i=? 5. 3767 X 8|=? 14. 179 X 2-i=? 6. 9987X 25^? 15. 769 X 3i=? 7. 9174X125 =? 16. 12-' X 19 =? 8. 3689X 33^ = ? 17. 125^X787 =? 9. 9210X 16'j=? 18. 250 X125 =? 10. 7897X166^=? 19. 16* X 48 =? 11. 8997X 50 =? 20. 83^x756 =? 12. 786X 14$=? 21. 197 X 12=? PROPERTIES OF NUMBERS. 81 23. 675 yards @ 37^ cents. OPERATION. 675 at & dollar $675.00 at 25 c=J "168/75" at 12^ c^J- 84.374 ^w. 253^121 24. 715X62J cents. 31. 9876x$2.18f 25. 947XS7A 32. 719x?3.62 26. 194xl8f 33. 965x$4.37J 27. 567X31} 34. 758x$l-25~ 28. 619X374 35. 197x$2.87.V 29. 1060X324 36. 879x$3.95~ 30. 197X75" 37. 179XW.32J 38. To find the cost where there are fractions in both factors: 18| Ibs. @ 12i cents. OPERATION. 18| Ibs. @ $1=$18.75 at 12in=^ or $2.34| Arts. 39. 37J Ibs. @ 18| cents. OPERATION. 37^ @ $1=837.50 at 12^=-^ 4.687 at 6^=4 2.343 Am. $7.030 40. 176^ doz. @ 18| cents. 41. 164J Ibs. @ $1.62J 73l| doz. @ 12J 374J Ibs. @ 2.25 doz. @ 37J 693J Ibs. @ 0.16f Amount, $197.06. Amount, $1224.93. ' E . The multiplier in the 25th Ex. wants only 12J cents, or J, of being a dollar; so we find the cost of 947 at a dollar and take off I of it. For 32J take 30, and 2J as J- of 10. ill /rf 82 NELSON'S COMMON-SCHOOL ARITHMETIC. IX. EASY FRACTIONS.* 67. A FRACTION is a part or number of parts of any thing considered as a whole. Fractions are of two kinds, common and decimal. A common fraction is written with two numbers, called terms, having a line between them, as j^; a decimal fraction with one number, having a period at the left, as .5 (five-tenths). 68. A common fraction indicates division, the upper number being the dividend and the lower the divisor. In treating of fractions, the dividend is called the numerator and the divisor the denominator. The denominator indicates the number of parts into which the whole is divided, and the numerator the num- ber of such parts under consideration. 69. VALUE OF A FRACTION. The lowest value of a fraction is expressed by the figure 1 for a numerator, and the highest value a number as great as the denominator less l.f ^ represents the lowest value of fractions of the denomination of ninths, while . | represents the highest value of that denomination. J *This chapter is introduced for the benefit of that large class of pcholars who leave school before completing the study of Arith- metic. The subject of fractions is treated of at length in the latter part of this book. |This does not apply to improper fractions, which, as the name indicates, are not strictly fractions. 1. Since this is the case, it is evident that fractions decrease in value as their denominators increase, the numerators remaining the same. 1 is less than ^, 1 than 1, than 1. 2. It is also evident that the value of a fraction depends on the EASY FRACTIONS. 83 When a number is divided into two parts, each part is called a half ; into 3 parts, each part is called a third; into 4 parts, each part is called a fourth; into 5, a fifth; into 12, a twelfth; into 18, an eighteenth; into 25, a twen- ty-fifth; into 100, a hundredth; into 476, a four hundred and seventy-sixth part. ORAL EXERCISES. 1. When a number is divided into 10 parts, what is each part called? Into 11? Into 20? Into 33? Into 45? Into 97? Into 62? 2. When divided into 31, what? Into 69? Into 103? Into 364? Into 155? Into 1000? Into 3144? 3. Which is the greater fraction, ^ or J ? -| or J? o'i? TYr T ' s ? 2 'ar ,',? ,' s or &? 4. Which is greater, f or |? Ans. |. REASON. Because it will take less to make it a whole number. The first fraction requires J to make it a whole number, while this one requires only J. 5. Which is the greater, f or f ? |orf? |or|? f or I? f or I? ?or T y if or ji? f or ?? 6. Which is the greater, if or ff ? ^f or f|? jf or H? i*orf? IforAf? 14 or if? Since the value of a fraction depends upon the relation of the numerator to the denominator, [note 2, page 82,] both terms may be multiplied or divided by the same number without altering its value. _4 ?^- 2 _ 1 ~ ~~ relation of the numerator to the denominator, or, in other words, the number of times the numerator is contained in the denomina- tor. | is equal to |, because the numerator 3 is contained in its denominator, 6, the same number of times that the numerator 4 ia contained in the denominator 8. 4 NELSON'S COMMON-SCHOOL ARITHMETIC. Now, | and i possess the same value as f , because their respective numerators are contained the same num- ber of times in their denominators. EXERCISES FOR THE SLATE. 5. Change | to twentieths. 3X& 15 EXPLANATION. By multiplying the 4 by 5, we change 4X& ^0 the denominator to twentieths; arid by multiplying the numerator by the same number we preserve the same value. 1. Change -| to 8ths; to 12ths; 4 to 20ths; f to Uths; I to 12ths; f to 18ths; to BOths. 2. Change ^ to 20ths; f to IGths; f to 27ths; j| to 52ds; | to 25ths; fg to 150ths. 3. Change | to 32ds; g to 40ths; Jf to 72ds; fg to 104ths. 4. Change to SOths; if to 52ds; || to 128ths; T 4 to 50ths. 5. Change T ^ to 6ths ; ^ to 5ths; T 8 2 to 4ths; j| to 8ths; if to 7ths; j to 8ths. 6. Change | to halves; T 6 a to 5ths; T 4 ^ to 4ths; || to 8ths; fg to 3ds. The fractions in last exercise, (6th,) when changed as required,' would be reduced to their lowest terms; that is, expressed in their simplest form. 70. To reduce a fraction to its lowest terms is, there- fore, to divide the numerator and denominator by such a number or numbers as will do so without a remainder. When the terms can not be divided exactly by any num- ber greater than 1, the fraction is in its simplest form. Reduce T 6 2 to its lowest terms. T y, T? , , T | g , ^ -fa to their lowest terms, Answers: , , T ' r , &, i, 4, ^. EASY FRACTIONS. 85 When a single number will not reduce the fraction, other numbers may be used, as below. 2. Reduce --f f-g- to its lowest terms. 3. Reduce to their lowest terms, -%, -f^, -J|f, and 4. Reduce to their lowest terms, -$> TT^ sWb^ T5~3T' 5. Reduce to their lowest terms, ^ 7 e 4 7 %, ^Irolb anc ^ Answers : ff, ffr, A. H- A> A. Mi. f, T&> #7, TTHF5 ITT' "8"' Fractions may be Proper, Improper, Simple, Compound or Complex. We shall treat of only the three former at present. A proper fraction is one whose numerator is less than its denominator, as ^. An improper fraction is one whose numerator is equal to or greater than its denominator, nc 5 P as TJ, TJ. 71. A simple fraction is a single fraction, and may be proper or improper, as |, |. 72. When a whole number and fraction appear together, they are called a mixed number, as 5|. 73. Improper fractions may be changed to whole or mixed numbers by dividing the numerator by the denominator.* To change - 1 /- to a mixed number. 5)13 EXPLANATION. There are 5 fifths in one whole num- ~T 3 ber; in 13 fifths there are as many Is as the number of 5 times 5 is contained in 13, which is two times, with 3 fifths over, making 2^. *This is simply acting on the principle that the numerator is the dividend and the denominator the divisor. 86 NELSON'S COMMON-SCHOOL ARITHMETIC. 1. Change |, g, |, J g &, 2.^-L, *. to whole or mixed num- bers. 2. Change the following : -*/, ^_, *-, - 2 T 2 -, ^-, -^-, 1. 3. Change i-f o, ^-, -^ ff, 44,1, ^, A J a - Answers: lj, If, 3, H, 301, 691, 53,. 8, 811, 7, 58^, 56f, 184, 13|, 3^, 81, Hi, 4i, 91, 18f . 74. To change whole or mixed numbers to improper frac- tions is an operation the reverse of the last, which, scarcely needs explanation. 4. Change 94 to an improper fraction. 9i EXPLANATION. In 1 whole number there are 5 fifths; 5 in 9 there are 9 times 5 or 5 times 9 fifths, to which we "JjJ add 4 fifths, and we have 4 ^. 5 5. * Change the following mixed numbers to improper fractions: 3|, 9|, 8?, 5|, 41f, 97J-, 16|. 6. Change the following: 7|, 9}J, 4|, 7 If, 18|, 16, 124. 7. Change the following: 21 to fifths; 16| to eighths; 121 to fourths; 16 to twelfths; 8 to twelfths; 131 to sixteenths. 75. To multiply a fraction by a whole number is simply to multiply the numerator without altering the denomi- nator, or to divide the denominator without altering the numerator. To multiply T 7 2 by 6. REASON. Assuming that 7 is a whole number, multiplying it by 6 gives 42; but since it is not a whole number, but twelfths, the 42 is =8- or 3. *The learner should prove the accuracy of his work by last ar- ticle. EASY FRACTIONS. 87 2- 6)/ 3 (.J=3J. By decreasing the denominator, the fraction is in- creased (as it takes fewer of the small parts to make a whole number) ; hence, the 7 represents halves instead of twelfths. =3J. 1. | X7=? 5. J> u xl2=? 9. i*xll=? 2. I X9=? 6. f|x 6? 10. f|Xl2=? 3. 5x8=? 7. fix 5-=? 11. Jfx 8=? 4. T * B X4=? 8. i|XH=? 12. X21=? Answers: l-, 6|, 5, 5f, 10|, 9&, 7|, 4ft, 4j, 6, 1||, 12|, 6. 76. To multiply a whole number by a fraction, we mul- tiply the numerator without altering the denominator. 13. Multiply 25 by |. 25X3 fourths=75 fourths, or 7 ? 5 , which, changed to a mixed number, [Art. 73]=18j. 14. 35X|^-? 18. 134X 2 1 o-=? 22. 16X T 3 o= ? 15. 21xf=? 19. 215X1 ? 23. 21X|=? 16. 18X^=? 20.112X|=? 24. 17. 116XA=? 21. 36X|=? 25. Answers: 28, 6/ ff , 4|, 174, 95|, 18, 3|, 931, 12, i|, 20, 34|, 30|, 77. To multiply a mixed number by a whole number. Multiply 7| by 9. 4 EXPLANATION. 3 fourths multiplied by 9=27 fourths, 9 or 6|; and the 7 multiplied by nme=63, plus the 669, 69^ making the product 69|. Or thus: 7| The Teacher will find it important to require the learner to preserve the process, as he will be apt to adopt clumsy methods of solution. 88 NELSON'S COMMON-SCHOOL ARITHMETIC. 2i?. 1SJX 5=? 30. 29JX 8=? 34. S3 i X 7=? 27. :>:\\ 8=? 31. IG-x 9= ? 35. 12^ X 8=? 28. 12-.i\12^=? :>- 8t|xl2=? 36. 5.,- g X 9? 21). :^\ 9=? 33. 62JX 9=? 37. 187 XH=? Answers: 3!, 93J, 3tf 62, 45 ,V 96 : -. i;>>. 1050, i;>o. ;u>o." 78. ^' multiply a whole number by a jni.ird number. 38. Multiply 29 by 8|. 29 S; FMM. vNvriox. Multiplying -9 by i! thirds, we hare TIT 58 thirxls. or l^J, which we write in the first line. Then *Z* 29X8=232, which, added to 1%=251 j. Or thus : 29X-V= 1 3 d =- 51 ] 15X 3]=? 42. 12\12i^? 4. r i. 14xl7f=zT 40 27X 6|=? 43.47X37^=? 46. 29x18 1 =? 41. ? 44. 93X16J^? 47. 83x 6 T " Ai> ' : . L-2J. 543|, 522^ 150. 17t52^." To multiply a fraction by a fraction. 4$. Multiply j by j. Assuming the numerator 5 to be ft whole number, jx^-^^ 1 - 5 : but 5 is not ft whole number, but 5 sixths; hence y is 6 timt* too MK& tj* dirided by 6z=|| T or | [Note l r p*ge 82.] 79. Hence, to multiply & fraction by a fraction, we iHufcyrfjr the tttraierafer* together for a new numerator, and the denominators for a new denominator. ? which, reduced to its lowest *It will be obserred that to mnltiplj by ft fraction does not m- II Mini Ike ultipli^^ as in whole numbers; bat, on the contrary, Aerates it, the f being less ihan J. To account for this, U Ls only necessary to remember that ft whole EASY FRACTIONS. 49. -| X 2=? 52. X= ? 5J. !;!xW ? 54 - *Xj=? 57. g/. ?=? Answers: f, , J, f, T ^, 3$, J, f, |, ,', 80. ^ multiply a mixed number by a fraction or a miw.d number. 58. Multiply 15| by T V 15|=^, which, multiplied by f =* 4 J- or 14 4 7 . 59. Multiply 8} by Ifif. 8|=3/t an(1 lf; 3=^- 5 4 4 X^ 60. 12^/lf^ -='! W>. 14'|X i 9 =? 66. 61. 8JX2%=? 64. 23^X i=? 67. 62. 37^X52*=? 65. |Xl4^^? 68. Answers: 126J, 35 , 231 J, 6 T 3 ^, ^, 208^, 245g, 29f, 1978f 81. To divide a wlwle number by a fraction or a mixed number. 1. Divide 315 by |, or, in other words, find how often | is contained in 315. .SOLUTION. Before we can measure 315 by fourths, we must change it to fourths. In 1 there are 4 fourths; in 315 there are 315 times 4 or 1260 fourths, which, divided by 3^420. Hence, \ is t contained in 315 420 times. OPERATION. 315 or 4 3)1260 430 t number is reduced to the denomination of a fraction by being mul- tiplied by it. 6Xf =18 fifths or 3|. Much more is a fraction re- duced in value if multiplied by a fraction. From this we readily infer, 2. That to divide by a fraction increases the dividend. 8 90 NELSON'S COMMON-SCHOOL ARITHMETIC. 136-5-?=? 3. 27-f-J=? 5. 684-f- T =? 7. Answers: 32|, 365|, 24341, 19740, 2280, 158f, 24. 8. Divide 25 by 5 J. OPERATION. 25x2 halves 5 ^ and 5|X2 ty. 50-f- 82. Hence, to divide by a fraction, we multiply by the denominator and divide by the numerator, or invert the divisor and proceed as in multiplication. 9. 157-4- 3|=? 12. 345-4- 6J=? 15. 195-=-16f=? 10. 22-5-12=? 13. 39-4-15I-? 16. 39-5-12f=? 11. 16-5-16|=? 14. 79-4-37^? 17. 87-4-3l|=? To divide one fraction by another. 18. Divide f by |. OPERATION. f Xf ig T 9 o EXPLANATION. By inverting the divisor, we obtain ^g, the terms of which, being divided by 2, give T 9 . 19. 9 _!_!_-? 22. 20. /,-*-=? 23. 21. 2_j_6 = ? 24. Answers : 1|, ff , f|, 38|f, 36|, 83. To divide when either divisor or dividend is a mixed number and the other term a whole number, both terms may be reduced to the same denomination. [Art. 81.] 28. Divide 3457] by 13. EXPLANATION. The dividend containing the fraction of ^, both terms are reduced to fourths, and division performed as in whole numbers. The result shows that the divisor is contained in the dividend 265 times, with a remainder of 49 fourths [Art. 46], or 265|| times. -4-=? 25. -f-l= -4- =? . 26. 27. -4-6|=? f, f 3457J 4 52)13829(265 104 342 312 309 260 49 EASY FRACTIONS. 91 The same by short division. 13)3457 1 EXPLANATION. 13 is contained in 8457, 265 times, ope 4 9 with a remainder of 12, which, reduced to fourths, 52 including the |- in the dividend, is 49 fourths. 13 not being contained in this an even number of times, the denomi- nator is increased 13 times, (which is the same as to decrease the numerator,) which gives the same fraction as by long division, 4| 29. 1398J 56 =? 35. 1255 | 350=? 30. 256J-- 7 =? 36. 796 J 421=? 31. 1939 -~ %=? 37. 467 f 12=? 32. 7961?-f-300 =? 38. 214 1 9=? 33. 9219 ~ 6J=? 39. 713^ 8=? 34. 1391 -v- 56=? 40. 391Jt 6=? Answers: 24}g|, 24^, 1475&, 36 T \, 232^, 26ffg, 84. To subtract a fraction from another of the same de- nomination is simply to subtract the less numerator from the greater. I. From T 7 a take T 3 . T 7 T 3 0=T 4 r I 2- jf-H=? 5. T - T V=? 8. T g -^=? 337 '25 _ ? (\ 1:2 _ 9 _ 9 Q 1 ( > 3 _ 9 ' 42 - 42 - ' ' 3T - 51 *' 3^ - 39 - r 416 5 _ ? 7 16 7 _ 9 10 11 _ 6 _ 9 55 - 52 - ' f' 45 - 45 - f 1U ' T7 IT - ' Answers: f, 4, J, T \, J, J, |, |, T 6 S , T \. 85. ^> subtract a fraction or mixed nnmber from a whole number. II. From 9 take 3|. The following formula will render the operation simple : Whole number. Fourths. EXPLANATION. Arranging the less under the greater, we find we can not take 3 fourths _ from fourths; so a whole number or 1 is 5 1 added to both terms. In 1 there are 4 fourths, QT 5i from which take 3 fourths, and we have a remainder of 1 fourth. To the 3 add 1 and 92 NELSON'S COMMON-SCHOOL ARITHMETIC. we have 4, which, subtracted from 9, leaves 5, giving for the an- swer 5. 12. 13 4J=? 15. 11 2 J=? 18. 52 2?4=? 13. 15 5=? 16. 7 f =? 19. 13 12J=? 14. 29121=? 17. 14 1 T 3 2 =? 20. 89 75J=? A n <JWAra 5 1 O5 J.1 94.6 Q2 Q3 11 1Q3 1Q3 Answers. g , o g , J g , *^, *^, o 3 , p|, 10^, io ? , lo^. 86. j?b subtract one fraction from another of a different denomination, it will be necessary first to reduce both to the same or a common denominator. 21. From | take f By Art. 69, | can be changed to 56ths by multiplying both terms by 7, and f can be changed to 56ths by mul- tiplying both terms by 8, giving || and |-|, the difference between which is ^, the answer. It will be observed that the multipliers used in this case were the two denominators, 7 and 8, which, multiplied to- gether, give a common denominator, and multiplied into the numerators of each other give the new numerators. OPERATION. | =ff-4|= 5 i ff 22. From f take i . 25. | 4=? 28. 23. From f take \ . 26. 6J 1=? 29. 24. From f take T 3 . 27. l| J= ? 30. T 3_ J_^? Answers: 2 5 , 5|, |f, |, T %, J, f|, |, ^, |. 87 ^ -c?cZ fractions of the same denomination, the nu- merators only are added, and the sum reduced to a mixed number or its lowest terms. 1. Add I+I+I+I- 3 EXPLANATION. Here the four numerators are added 6 together, making 21 eighths, which, reduced to a mixed 5 number, are equal to 2. \ EASY FRACTIONS. 93 2- 3 31 4 1718 9~r y ~r$ i 3 4. 5- 6- 88. ^> add fractions of different denominations, they should first be reduced to a common denominator, as iu subtraction. 7. |+!-? |+|=|+|=Vt=lf or If When three or more fractions of different denomina- tions are to be added together, they may be reduced to a common denominator by multiplying all the denominators together, as above, and then by multiplying each numer- ator by all the denominators except its own.* 8. Find the sum of i+f + 2X^X6 48=: Common denominator. 1 X 4 X 6==24=sJ'Wfcl numerator. 3x2X^36 Second numerator. 5x2x4 40 ^/ureZ numerator. Wfa^Stim of numerators. Hence, ^=2^=2^. The J in the example was multiplied by 24, giving ||; the f by 12, giving || ; and the | by 8, giving |g. 9. + =? 14. | + | + | -? 19. 2|+ | + | =? 10. i + 2=? 15. i + | + i=? 20.5-i+6|+|=? 11-T\+I= ? 16 - 7VH+.iV= ? 21. i+2|+ T 3 2 =? 12. 2 + |=? 17. |+V 3 +'l=? 22. |+ + -^=? 13. J + T V=? 18. | + I +/ 3 =? 23. ^+ ^+ J =? Answers: 2^, 1, 1^, Ii, **, i 12, 3|, 1|, 1 I/,, *This is simply multiplying both terms by the same number. [Art. 69.] 94 NELSON'S COMMON-SCHOOL ARITHMETIC. X. THE MERCANTILE PROFESSION". THE mercantile community may be divided into various classes : Importers, Jobbers, Wholesale Dealers, Commis- sion, Forwarding and Retail Merchants, Brokers, etc. 89. Importers purchase goods and produce in foreign countries, and sell them in the home market to jobbers and wholesale dealers. They also receive goods from abroad to sell on commission. 90. Jobbers. This term was first applied to persons dealing in stocks to a limited extent, but it now includes nearly all classes of wholesale dealers. We speak of dry goods jobbers, produce jobbers, cattle jobbers, etc. 91. Wholesale Merchants or dealers buy from import- ers, jobbers, manufacturers and producers, and sell to re- tail dealers. 92. Commission Merchants* act as agents for other per- sons in buying and selling goods, collecting debts, etc., for which they charge a percentage on the whole amount of sale, purchase or collection. Merchants of this class usually keep a wholesale department in their warehouses, where they sell their own goods as well as those of others, and even ship merchandise to distant places for sale on commission, thus acting in the capacity of principals as well as agents. The person who sends goods to another to be sold on com- mission is called the shipper or consignor ; the person who receives them, agent, correspondent, consignee or factor ; and the goods or merchandise sent, shipment or consignment. 93. Forwarding Commission Merchants and Express Companies are intrusted with the care of conveying goods * Auctioneers belong to this class. THE MERCANTILE PROFESSION, 95 from one city or country to another, for which they charge a percentage, called forwarding commission. This class of merchants usually have warehouses, and on rivers, wharf-boats, for the storage of merchandise. A separate charge is made for all goods lodged in these places, ac- cording to the time they remain. The fee charged is called storage. 94. Retail Merchants dispose of their goods in small quantities suited to the wants of consumers. 95. Brokers form a numerous class. They assist com- mission merchants and dealers generally in finding buyers for their wares, and trade or speculate in stocks, lands, etc. By confining their attention to one line of business, they acquire a more intimate knowledge of its details and of the credit of persons engaged in it, and are thus pre- pared to render valuable services to both buyer and seller, between whom they act as middle-men. The business of the broker does not require the invest- ment of much capital, as, unlike commission merchants, they are not under the necessity of keeping stores or warehouses. For the same reason, their fees are smaller than those of the latter. There are Money, Exchange and Bill, Stock, Custom- house, Heal Estate and other Brokers. MONEY, EXCHANGE AND BILL BROKERS. 96. They buy, sell and exchange specie, bills of ex- change, notes, etc. The entire business of this class is often performed by one individual or company. Being considered a branch of the banking business, many of them adopt the name of bankers. Heal Estate and Stock Brokers buy and sell for others, 9G NELSON'S COMMON-SCHOOL ARITHMETIC. lands, houses, stocks in public funds and joint stock com- panies, etc. Custom-house Brokers find employment in maritime cities, by assisting masters of ships in obtaining the nec- essary papers at the custom-house, and paying duties or taxes incident to the navigation of the ocean. Ship and Insurance Brokers procure freights and car- goes for ships, adjust the terms of charter parties, settle accounts between owners and masters of ships, effect in- surance on ships and cargoes, etc. Produce Brokers buy and sell for others various kinds of farm produce, as corn, wheat, cheese, etc. They stand between the producer and dealer or shipper. The business of other brokers will be sufficiently indi- cated by their names. MERCANTILE AND COMMERCIAL COLLEGES are institutions of learning, having for their ostensible ob- ject to prepare young men for entering the mercantile profession. They are got up by private enterprise, some of them being chartered and some not. The chartered institutions possess no advantages over the others, as none of them have the power of conferring degrees. The course of study in this class of colleges usually comprises instruction in book-keeping, with its application to the various branches of trade, manufactures, etc.; mer- cantile arithmetic, penmanship and business correspond- ence ; lectures on the usages of trade, negotiation of business paper and the most useful branches of commer- cial law. When conducted with ability and integrity, commercial educational establishments rank among the most useful institutions of learning of the day. Though of compara- THE MERCANTILE PROFESSION. 97 tively recent origin, they are to be found in most of the larger cities of the Union, and receive a liberal patronage by all classes of the community. Professional men, me- chanics, farmers, and, in large cities, ladies, are to be found among the number who consider their education unfinished until they have passed a commercial course in one of these, and merited a diploma. It may be safely asserted that in no institution of learn- ing is there as much useful information obtained in so short a time, and at such trifling expense, as in commercial colleges. Business men may obtain a knowledge of book-keeping, as applied to their own business, in .a few weeks, while most youths might be profitably engaged in a college at least one year. PERSONS EMPLOYED IN BUSINESS HOUSES. The persons employed in mercantile houses are: Book- keepers, Correspondents, Solicitors, Salesmen, Travelers; Entry, Bill, Shipping and Engrossing Clerks, Junior Clerks or Boys, Porters, Coopers and Draymen. DUTIES OF THE VARIOUS OFFICES. Book-keeper. The book-keeper's place of business is in the counting-room. His duty is to keep the accounts of the establishment, in a variety of books for that purpose, to receive and pay out all moneys, and deposit money in the banks for safe keeping, to make out bills, render ac- counts, statements, etc., from the ledger and sometimes to conduct the business correspondence. Correspondent. The business of the correspondent is to reply to letters of inquiry, and to write all letters of bus- iness connected with the house, etc. In extensive im- 9 98 NELSON'S COMMON-SCHOOL ARITHMETIC. porting houses, the correspondent is usually a person who understands some three or four of the modern languages. The Second Boole-keeper assists the first book-keeper. He copies or transfers to a journal or day-book the items found in the sales-book, journalizes and posts to the ledger. It devolves upon him to make out bills, accounts, and to assist in the counting-room generally. The Solicitor traveling agent belongs to the broker's class. His business is to solicit orders and secure buyers for houses with which he has made previous arrangements. Accordingly he is found in the hotels, on the steamboats or at the railroad depots. When he finds a buyer, he con- ducts him to the store for which he is operating, and, if not under salary, receives a commission for his services. Clerk is a general term applied to all employes, the porter, drayman and cooper excepted. Salesman. The duties of the salesman consist in un- packing, marking and arranging goods for sale, receiving customers and selling to them. In some houses, the salesman receives a commission for the amount of trade he influences, and in all places the amount of his salary very much depends upon this cir- cumstance. These statements apply principally to wholesale busi- ness. The Skipping Clerk receives and examines goods to see if they agree with the conditions of the bill of lading, and attends to the shipping of goods from the establishment. These he enters in a book for that purpose, called the shipping-book. The Entry Clerk records the sales made by the sales- man in a book called a blotter or sales-book. The Bill Clerk makes out the bills or outward invoices from the sales- book. THE MERCANTILE PROFESSION. 99 Entry and bill clerks should be rapid penmen and ex- pert in figures, if they would command liberal salaries. The Engrossing Cleric assists generally, sometimes in the counting-room, but more generally as entry or bill clerk. He is simply a copyist.* Junior Clerks are usually boys from 12 to 17 years of age, whose duty it is to run on errands, pack and unpack goods, mark boxes and packages, and keep sale-rooms in, order. After acting in the capacity of junior clerk for two or three years, they are promoted to more lucrative and re- sponsible offices. The youth who would aspire to a high degree of use- fulness~in his profession, should not rest content with the I / 1 opportunities for improvement afforded by the store or * j J sales-room. His evenings should be devoted to useful study, until he acquires proficiency in all the various branches of mercantile business. The study of Freedley's excellent " Treatise on Busi- ness" is highly recommended, especially chap, iii, page 46. Heading-rooms, libraries, mercantile associations, mer- cantile and commercial colleges are to be found in most of the larger cities of the Union, and should form the places of resort for young men of this class in preference to questionable places of amusement, too much frequented by them. Porter. The business of the porter is to open and close the store, keep the store and counting-rooms clean and in order, pack and unpack goods, assist in handling and weighing heavy goods, marking packages, etc. ~*In many houses, the whole business of clerking is performed by one person, while in others many more offices are called into requi- sition than are noticed here. r '// m 100 NELSON'S COMMON-SCHOOL ARITHMETIC. The office of porter is a more responsible one than most people imagine. By the faithful discharge of its duties, hundreds of men in this country have been placed in pos- session of respectable retail houses. In some large establishments, there are two or more porters engaged, between whom the duties of the above are divided. Cooper. In liquor and heavy sugar establishments, pack- ing-houses, etc., the services of a cooper are required, whose duty it is to open and close hogsheads, barrels, etc., and to repair damages to which such articles are subject in carrying. Drayman. The drayman acts as "carrier" between the etore and depot, landing or wharf. He usually keeps a book called a dray-book, in which are entered the con- tents of each load. This is signed by the clerk at the place of delivery, and when the entire shipment is made the amount is entered in the Bill of Lading. XL BILLS INVOICES. 97. WHEN goods are sold, it is the duty of the mer- chant, or one of his clerks, to make out a statement of ] the quantity, kind and price of each article, for the sat- \ isfaction of the purchaser, and to enter at the foot of such > statement the whole amount of the purchase, with the ; payment received, if any, or the terms of settlement. If the goods are bought to sell again, this statement is com- monly called an Invoice; otherwise it is called a Bill, es- pecially by the purchaser. A bill or invoice is sometimes delivered to the buyer at; the time of purchase; but it is usually sent with the goods, or, if the buyer resides at a distance, by mail. An invoice should specify the place and date of sale, THE MERCANTILE PROFESSION. 101 the name of buyer and seller, a description of the goods, the price of boxes, etc., used for packing, charges for in- surance, and, when 'payment is not made, the terms of sale. When goods are received, the quality and quantity are compared with the invoice, and the selling prices made out from it, after which it is filed away or pasted in a book for future reference. 98. It is the custom of merchants to have their bill heads (heading of bills) printed, with names of city and street, number of house, and such other matter as will fa- cilitate the labor of clerks, or otherwise advance the inter- ests of the business. A specimen form of such heading will be given in the bills that follow. 99. Filing Bills. When there are many bills on hand designed for collection, they should be folded neatly of the same length and breadth, and have the names, addresses and amounts written on the outside at the top. A gum band will then keep them firmly in their places, and per- mit their being delivered without the trouble of opening for examination. 100. Retail bills are rendered periodically, by the month, quarter, half-year or year, according to agreement or the usage of the house. When a pass-book is not kept, it is well to have a memorandum of each purchase, so that in rendering his final bill the merchant need not insert the items. 101. Account or Statement. The final bill of a mer- ! chant now goes by the name of account or statement. The ! head contains the date upon which it is drawn, and the ! word "To" substituted for "Bought of." In the margin, I on the left, are the dates of the several purchases, with tho | words, "For amt. rendered," or "Amt. pr. bill rendered." 102. In making out bills, the three requisites are ra- j pidity, legibility and accuracy. The principal is accuracy. 102 NELSON'S COMMON-SCHOOL ARITHMETIC. In business, it is not enough to be right after one or two, or perhaps repeated, attempts. The clerk should be cor- rect the first attempt, and generally is so. Boys designed for business pursuits ought to spend much time at bill- making, until they acquire familiarity with the numerous abbreviations, and can make out a bill from dictation al- almost as rapidly as the items can be called off. 103. Finding the cost of a number of articles at a cer- tain price, and placing the amount opposite, is called, in bill-making, extending; adding the columns, footing up. 104. Receipt on a Bill. A clerk or agent may write the name of his employer to a receipt and it will be good, if he write his own initials or last name underneath. BOOKSELLERS AND STATIONERS' BILLS. CINCINNATI, June 16, 1866. MR. HORATIO NELSON: Bought of R. W. CARROLL & CO., PUBLISHERS, BOOKSELLERS & STATIONERS, WHOLESALE & RETAIL, 117 WEST FOURTH STREET. TERMS : In making orders, be particular to avoid mistakes. All claims for Errors or Damages to be made within five days of receipt of goods 1 Gro. Pen-holders, 2 35 *[ u u u 75 3 Doz. Paper-Cutters, .... 2 " Ebony Rulers, .... 1 Rm. Bill Cap, 2.00 3.75 6 7 7 00 50 00 2 Letter Cap, No. 1400, . . 1 2-Qr. Blank Book, .... 3 Qr. Blotting Paper, .... 8.25 40 2.25 16 6 50 80 75 $47.55 Rec'd Paym't. R. W. CARROLL & Co. DAVIS. Gro., Gross; Doz., Dozen; Qr., Quire; Rm., Ream. 4 BILLSINVOICES. 103 BOOKS STATIONERY. 105. Books, stationery, etc., at wholesale, are usually sold by the dozen. Paper is sold by the ream, bundle or pound. Writing paper is put up in half-reams; printing paper in bundles of two reams. In the exercises which follow, the teacher will find it to the advantage of scholars to have them write from dicta- tion. The bills may be made out in favor of the learner or otherwise. 3. J dz. Hooker's Nat. Philos., $14.40; 1 rm. cap. $6.50; 1 do. bill, $7.00; 2 dz.!2-in. ebony rulers, $3.75; 2 dz. paper- cutters, $2.00; 1 bx. crayons, 35 c; 12 dz. pass-books, 40 c 4. 2 dz. mucilage, $2.75; 1 dz. carmine, $2.00; 2 dz. tin cutters, $2.00; 2 dz. rulers, $4.50; -J- rm. natl. cap, $5.50; 3 dz. No. 4 pass-bks 44 c rm. treas'y cap, $8.40; 1 qr. blotting, $2.25. 5. 1 bx. 4560 5^ envelopes, $2.25; 1 do. 8J Manilla, $2.75; 1 dz. Lincoln portraits, $3.00; 2 dz. check-books, $12.24; 1000 penman's blanks, $20.50; 21 T 9 3 dz. bill- books, $2.16; 4 dz. do. 2 qr. ea., $2.50; 20 dz. check, $2.16; 20 dz. inv., $1.00; 20 dz. day-books, $1.80; 60 dz. journals, $1.80; 20 dz. ledgers, $1.80; sunds,, $7.00. Credit cash, $210. 6. 1 copy'g press, $10.00; | dz. Rec. of a Country Par- son, $20; T V dz. Wordsworth's Poems, $25.00; T \ dz. Be- ginning Life, $12.50; y 5 ^ dz. Heads and Hands in the Wrld of Labor, $12.00; i Essay on Woman's Work, 12.50; 2 T % do. Nat'n'l Lyrics, $4.80. Answers: $55.80, $30.21, $33.75, $31.02, $150.16. SHOE BUSINESS. 106. Shoes are usually sold at wholesale by the dozen, boxes furnished gratuitously. 104 NELSON'S COMMON-SCHOOL ARITHMETIC. Pairs. 7. 12 Wos. Goat Tips Bals 1.45 6 Miss " " " 1.75 12 Childs " " 1.05 12 Calf " 1.05 12 Wos. Goat Tips " 2.50 10 " Last. Lace Cong 2.00 6 " But. Gait 2.90 6 Miss Kid Cong. " 1.90_ Numbers on boxes. Pairs. 8. S 12 Child Serge Lace Heel'd Gaiters. 65 474 13 Ladies Kid ch. nl. Balmorals 1.30 533 12 " Goat D. S. " 1.28 2386 11 Wos. Kid Cong. D. S. Boots.... 1.374 2449 12 " Serge Cong. Gaiters 1.20 2593 12 " " " " 1.124 2475 24 Child Gai. Peg-Heel'd J. L. Boots 45 2586 12 " End. u " cop. tip. 45 2580 24 " Buff " Lace " " 40 2575 12 Miss GaL " J. L 674 2413 15 " Goat " Cong Gait... 95 2517 12 " " " " . . . 60 2591 9 Ladies Goat, tip. ch. nl. Bals 2.00 2431 12 Youths Buff Peg-H'1'd " 724 2461 12 Ladies Calf " tip Bals.. 1.25 2588 12 Kid ch. H'l'd . . 1.50 2589 12 Peb. Calf ch. HTd tip Bals 1.50 2590 12 Miss Kid ScT. Welt " 1.05 ABBREVIATIONS. TFbs., Womans; D. 5., Double soled ; Cong., Con- gress; Grd.j Grained; End., Enameled; J. L., Jenny Lind; Cop* Tip., Copper tipped; Ch. NL, Channel nailed; Hid., Heeled; Sd, Wit., Sewed Welt; Peb. Cf., Pebbled calf; Bals., Balmorals; But. Oait., Button Gaiters. BILLS INVOICES. 105 MILLINERY. 9. 60 12 15-Braid Bonnets @ J0.62J 68 6 it 1 .25 70 4 7-Braid a u 1 .50 . 80 2 7 " a u 3 .00 86 2 7 " n u 3 .75 6 PCS. No. 1 Tafft. Ribbon a 15 5 a " 2 u u a 28 3 a " 4 u u u 48 2 u " 6 it it u 75 1 u " 12 u u tt 1 .10 3 tt Bonnet Ribbon u 2 .00 2 a u u a 2 .50 3 1 Box Ruches. . a 1 .50 415 1 ti u 2 .25 210 i Doz. Bunches Flowers u 18 .00 J " Feathers " 36.00 1 PC. Black Silk, 20 yds " 87J GROCERY BUSINESS. 107. TARES. In Cincinnati: New Orleans sugar, in hogsheads, 10%; Cuba and Porto Rico, 12%; sugar in boxes, 15%. Rice in tierces, 10%; indigo in ceroons, 11%; in boxes, actual tare; salt in barrels, each 30 Ibs.; coflee, cotton, spices, feathers and salt, in bags and bales, no tare; manufactured tobacco in kegs and boxes, (not enumerated,) actual tare; madder in casks, actual tare; lard and bacon in packages, actual tare. Lard kegs tared after being emptied.* * Butter in firkins is subject to an allowance of 2 Ibs. for soakage; in rolls packed in half barrels, 1 Ib.j barrels, 2 Ibs.; tubs 50 to 70 Ibs. 1 Ib. 106 NELSON'S COMMON-SCHOOL ARITHMETIC. In New York: The tares differ slightly from the above; "but the list being a long one, it is reserved for the revised edition of " Nelson's Mercantile Arithmetic." Extra Charges may be made for drayage, insurance, cooperage, storage, boxes, bags, etc., in which goods are packed in the store. No extra charge is made for the original package. In the bills which follow, some of these charges are introduced. 10. 1 Hhd. N. 0. Sugar, *^oo **1080 Ibs. .@$.07 4 Brls. N. 0. Molasses/H 169 gals ---- " 35 1 Trs. Eice, g 630 Ibs ............. " 4 20 Bags Rio Coffee, 3200 Ibs .......... " 11 2 Half Chests Black Tea,** o 1 ^** 72 Ibs. " 25 1 0028 3 " " Yng. Hyson do. 150 Ibs. . " 50 1 " " Imperial do. 60 Ibs.." 40 2 " " Gunpowder do. 110 Ibs.. " 60 1 " " Oolong Blk. Tea, 45 Ibs. " 40 VJ 1 Box tiAWAUVt. V^ A U.J-iC*il.iJ. \_fJLi 5 lump Tobacco, l l\ 108 Ibs.. cc 25 1 a pound lump " '4J 124 Ibs.. (C 20 1 u Va. pound " 4o 120 Ibs.. a 35 1 a 8 lump " J |g 125 Ibs.. u 22 20 Brls Rect. Whisky 800 gals a 17 4 a Ginger Wine, 160 gals a 60 \ Cask French Brandy, 40 gals a 4 .00 % a Port Wine, 45 gals .... a 2 00 10 Brls. Bourbon Whisky, 405 gals .... a 1 .00 J Brl. Holland Gin 20 gals.... u 1 .50 5 % * F , * Gross Weight. J Gallons in each barrel. {Tare, or weight of bag, box, etc. **Net Weight. BILLS-INVOICES. 107 The small figures on the left indicate the prices of boxes, barrels, etc. 11. 100 Boxes Cheese, 4 '?<> 3690 @$ .08 30 Firkins Butter, 3 f<j<> 2820 " 15 100 Boxes $2 Starch, 4810 " 5 100 " *** Star Candles, 4000 " 20 20 Bbls. $ * 5 Lard Oil, 810 gals " 85 50 " Mess Pork "16.00 10 Tierces S. C. C. Hams, 3 |f g 3000. " 11 30 Kegs Lard, 'J'g 1334 " 12 J 15 Bbls. Mess Beef 15.00 Com. for purchasing, $1521.75 " Z\% Drayages 16.00 Insurance on $5000 59.88 12. 1 bag Pepper, 103 10J 1 " Allspice, 128.. 10 4 dz. Shakers' Brooms 2.40 1 Em. Cap Paper 4.00 W% off. 1 " Med " 6.00 " " 1 " D. " 8.00 " " 5 bxs. 10 Ger. Ex. Soap, 297. 7 1 Keg Soda, 112 Ibs 5J 1 bx. 20 Saleratus, 61 Ibs.... 5J 1 " 20 Saltpeter, 47 u .... 9J Drayage 79 13. 10 Bbls, -.<> Sugar, 246 23 245 20 233 18 246 17 250 21 275 21 227 22 232 19 239 21 266 _25 2459 207=2252 lbs.@12|c, $ Drayage 1.60 108 NELSON'S COMMON-SCHOOL ARITHMETIC. 14. 1 Bbl. ' 35 Lard Oil, 41 85 1 Hf. bbl. 75 98% Alcohol, 21 54 1 Dz. Washboards 2.25 1 ' Sk. Bar Lead, 96 Ibs 6J 1 Pkg. Yarn, 28 Ibs 26 2 Doz. No. 2 Brooms . . . , 2.40 1 Trc. Ex. S. C. Drd. Beef 2 25 1 Ke<r lOd Nails 3.00 | Bbl. 5 Lard Oil, 23 95 Drayage 75 15. 2 Cases H. Cheese, 46 Ibs 28 4 " S. W. R. Cheese, ! ?| 112 Ibs.. 22 3 " E. D. " 87 78 u -fl 18 2 " N. W. R. " 2 5 21 " .. 17 1 Bbl. Cuba sugar, 2 |f 202 Ibs 13 J 4 Boxes MR. Raisins 4.40 2 " " " 2.40 2 i " " .. 1.45 3 Jars Prunes 45 5 Drums Figs, 37^ Ibs 24 1 Case 100 half boxes Sardines 42 1 Box American Castile Soap, 28 Ibs.. 16* 1 " French " " 35 " .. 16 1 Doz. Cox Sparkling Gelatine 6.00 2 Boxes * Saleratus, 128 Ibs 7^ 5 Half boxes Star Candles, 100 Ibs 24 1 Bbl. 25 Com. Smok. Tobacco, 100 Ibs 12 1 " 2S S. F. " " 83 " 25 2 Boxes 40 Starch, 96 Ibs 7 2 " Soda, 92 13 Sk., sack ; Pkg., package ; W. R., Western Reserve. 2 boxes raisins, 2 half boxes. BILLS INVOICES. 109 16. 1 Bbl. 25 Soft Refined Sugar, 240 Ibs. . @ 16 1 25 Hard " 220 .. " 17 1 " 25 Granulated " 212 " .. " 25 1 Sack Java Coffee, 136 Ibs " 41 1 " Rio " 156 " " 28 1 " Laguayra, 111 Ibs " 30 | Chest Y. H. Tea, 8214^68 Ibs... " 1.25 I " Black 53 12=41 Ibs... " 1.35 J " Y. H. Canton Tea, f * 68 Ibs. . . " 90 i " " Fine, ?o 66 Ibs "1.60 -J " Oolong Common Tea, }$ 41 Ibs. " 90 1 Sack African Peanuts, 2 bushels .... 2.40 1 " Roasted do., 3 bushels 3.10 1 Bbl. 25 Dried Peaches, } J 95 Ibs. . . 21 1 " 25 " Apples, 105 85 Ibs.... " 15 1 Box 20 German Soap, 64 Ibs . .. " 12 1 " 2 Common " 64 Ibs " 12 1 " 20 Cincinnati Extra do., 61 Ibs. . " 14 1 Bbl. * 5 Prm. Wh. Beans ~?j> 216 Ibs. " 2-f 2 Boxes 2 doz. Baking Powder " 4.80 17. 2 Sks. Rio Coffee, 323 25 1 Bbl. Molasses, 45 J 60 1 " 37 Ricej 23519 10 2 " 75 Sugar, iJl=?g 12 Drayage 50 DISCOUNT. An abatement entitled discount is often made on the bill for cash or when goods have fallen in price. When making such abatements, the clerk should remember that he is discounting the profits as well as the first cost. For instance, I buy goods at $100, and sell them at a profit of 50 per cent., which makes the price $150. Now, if this is discounted at 40 per cent., it does not follow that a gain of 10 per cent, is made. 40 per cent, of $150= $60, which, taken from the selling price, leaves $90, making, in- stead of a gain of 10 per cent., a loss to that amount. 110 NELSON'S COMMON-SCHOOL ARITHMETIC. 18. 20 Bbls. Molasses, 43J 44 43 42J 44J 411 44^ 43* 44 441 44 42^ 42J 43 43J 411 44 43 865J *2@75o 2.00 19. 10 Tcs. Shaffer's Hams 3 *'<> $0.21 5 " A. Spence's S. C. do. ^Jf 25 1 Bbl. 25 20 dz. Tongues 1.00 Com. on $1013.78 2%% Exch \% Drayage 2.00 Answers: $86.24, $262.26, $1046,23, $138.73, $285.60, $185.51, $4152.87, $1665.97, $649.63, $688./5, $131.90, $228.74, $130, $97.09. NOTE. When merchants can not fill the orders from their store, they sometimes buy elsewhere, and charge their customer cost and a commission, as in the last bill. The Teacher should not confine himself to the few exercises in bill-making given in this work, as he will find it both interesting and profitable to the learner to represent other kinds of business in the same way. Any number of bills can be dictated from price lists or prices current, to be had in any large city. * Leakage. "^ , BILLS INVOICES. HI DRY GOODS * Yds. Price 20. 1 PC. H. A. g- Blea. Muslin 45 25 1 " A | Light " 43 15 1 " D. Q. | Fine Bro. Muslin. . . . 43 22 2 " P. I Muslin 70 2 24 1 " L. Check 68 1 26 1 " W.Jeans 46 2 30 A3" Fey Prints 126 1 15 B 1 " " " 37 2 16 C 9 " " " 355 2 17 1 " BuffChambray 20 2 35 1 " Dom Gingham... 35 25 1 " L " 60 2 25 2 Doz. Coats' Spools 1.10 1 " | Hose 3.00 1 " Ladies' L. W. Hose 6.75 No.4 1 Pk. Pins 70 31" "..... 75 21" " 90 1 Doz. Lin. Hdkfs 6.00 1 " H. S. " 9.50 f 1 PC. Sheetgs to fill 35 2 24 2 Bxs 5 and Strapping 5 2.00 * Though practically correct, some of the answers to these bills will be found mathematically wrong. Accuracy in cents has been sacrificed in conformity with business usage, which often rejects fractions in extensions, alternately adding a cent and rejecting a half or fourth. The letters "H. A." etc., indicate the^rade of nius- liiis, or are the initials of the maker or factory ; the figures and letters in the margin are marked in the wholesale house to distin- guish lots of nearly the same kind from one another. tNot in the order; but will not be objected to by the buyer, as it prevents goods from shifting in the case. 112 NELSON'S COMMON-SCHOOL ARITHMETIC. 21. 4 PCS, , Prints . 185 3 17 1 tt Jeans . 45 1 30 1 a Cottonade . 38 1 37 s 5 it Chamb. Gingham . 154 2 35 2 it Luster . 101 2 30 2 Doz . Skirt Braid. 1 .00 10 u Coats' Spools 1 .10 No. 101 u Miss's Hoops 5.00 181 M a tt 9.00 201 u n it 10.00 30 4 a 2 Gore Miss's Hoops 2 16 .00 22. 1 PC. Fancy Cassimere 27f 2 .20 1 u French " 28| 2 .85 1 a Mixed " 31 2 .90 1 M Amoskeag A Ticks GO 47 1 U Glassgow Gingham 40J 27 1 u Victoria " 37 42 1 a Shawmut Can. Flannel .... 35J 34 2 3 u LX.L.Wl.do.41,64,34.. 139 73 1 u Roan. Striped Shirting .... 37J 31 2 a Fey Slk Dress Goods, jj.. 71 1 .10 1 tt Village Green Checks 37| 23 4 tt Lonsdale Checks f? ?<) 143 21 1 it Blk French Broadcloth . . . 29} 4. .25 1 tt English " 37J 5 .00 2 tt Bolivar Denims \ g 98 20* 1 Charter Oak Denims 60 22 1 a 38| 17 a 2 IK Duchess Delaines, 31 1 36 1 . . 67* 30 4 a Semper Idem, 40 38 27 1 36. 1151- 32 2 2 a Barre Brown Sheeting \ g . . 76 34 1 tt Great Falls do. do.. 42J 28 2.50 BILLS IN VOICES. 113 23. 75 48 Doz. Gent's Shakspeare Coll 2.00 6 60 u Ladies Plea " , 1.26 1 48 N " BvronEmb. " , 1.50 1 45 U " Gar. " " , 1.50 1 48 it u p er a tt 1.50 1 120 tt " Sq. Gar. C.&C.E. Coll. 1.00 1 29 a " Mull. Edge Gar, u 1.50 1 20 tt Vic. Cuffs, 2 C ., 2.50 1 2 a " ..Octagon C.&C.E .Sets 4.00 1 5 N " Sqr. Emb. u 5.50 1 8 (C " Point Wht. Trim. " 4.50 16 47 n Shirt Fronts 3.50 24. 11 10 PCS. 4 4 Shirting Linen 294 $0.37 12 10 a 4 4 tt it 297 40* 13 10 tt 4 it 300 44 14 15 (1 4 u 433 49 15 15 u A u u 447 55 16 10 tt 1 U It 285 60 17 10 4 tt U 287 85 25. 1342 2 Doz. Napkins 2.75 1343 2 u u 3.40 1356 2 a 4.50 1270 2 u it tl 4.00 1138 4 Felt Table Covers 2.85 1416 1 PC fl6 2 57* 1417 1 tt " " "3 26 62 2 1418 1 a " u " 4. . 26 70 1419 1 n tt u a /) 27 75 1421 1 u u a ^ 26 87 2 1422 1 tt a u a g 27 1.00 Emb., Embroidered; Gar., Garote; Per., Persiguey; C. $ 2 <?., Cord and 2 Cord; Vic., Victoria. 10 114 NELSON'S COMMON-SCHOOL ARITHMETIC. QUEENSWARE. MV 3 Doz. Edged Plates $0.40 10 " " " 50 5 " *CC " 50 " " Dishes, ea, $1.75, 2.25, 2.75. 1 " " Bakers, ea. $1.50,2.00, 2.50. 1 " " Beaded Nappies, ea. 1.75, 2.25 | " " Tureens 3.50 2 " " Bowls 80 2J " " " 60 3 " " " 50 \ " " Pitchers 3.50 1 " Colored " ea. $2.50, $4.00 4 " " Bowls 87 2 5 " " " 65 6 " " " 55 9 Sets CC Teas 20 36 " Painted Teas 20 Crate 1.00 27 HAT BUSINESS. \ Doz. Men's Black Cass. Hungarian 21.00 \ " " " " 27.00 6J " " " " 33.00 | " " " Broad Brim Wool. .. 14.00 1 " " Wove Senate 12.00 J " " " Cashmerette 15.00 \ " Boys' " Hungarian 7.00 4 " " Caps, assorted 12.00 i " " " " 9.75 J " " Cloth Caps 9.00 \ " Children's Fancy Caps 8.00 * CC, cream colored. i BILLS INVOICES. 115 28 HARDWARE. 25 Bars 1 JX J Bar Iron ......... 750 Ibs. $0.03 6 Bund. Round " ......... 625 " 4 4 11 x| Dandy Tire Iron.. 500 " 3 s 3 " 1 X| Horseshoe Iron^ 4 " 1 X T 7 e " " [ 975 " 4 2 " 1 X -\ " "3 2 Sets 1JX4 L. Pitts Springs^ 3 1JX5 " " " t- 500 10 2 " 2 v5 u " u 1 " 2 X8 ". " " 4 Slabs 14 X | Steel ........... .600 " 7 2 3 Doz. Amos No. 2 Shovels ............ 10.50 2 " Rowland's No. 2 Shovels ......... 8.50 1 " No. 2 Spades ......... 8.50 6 " No. 5684 Pocket Knives ......... 7.50 3 " " 4215 " " ....... 4.50 Answers: $238.88, $160.39, $56.73, $194.92, $1234.04 $832.60, $232.53, $981.51, $318.25, $735.63. MERCHANT TAILOR'S BILL. 9 - NEW YORK, Ap. 5, 1867. MR. A., To B. Jan. 3. For 1 Blue Beav. Overcoat 70.00 Feb. 9 1 Pr. Blk Doeskn Pants 23.50 $93.50 Mar. 2. Cr. By Cash 50.00 $43.50 To THE TEACHER. The learner ought to be taught that bills are seldom punctuated, and that even the dollar or cent signs or @ are rarely used. It is proper, however, to insert the dollar sign at foot- ings. 116 NELSON'S COMMON-SCHOOL ARITHMETIC. FARMER'S BILL WITH CREDITS. RICHMOND, Nov. 1, 1869. MESS. TRADER & Co., To JONATHAN FARMER. Jan. 3. For 3 Durham Heifers 25.00 75.00 " " 1 2-yr. old Bay Colt 50.00 Ap. 11. " 140 2 Bu. Corn 50 70.25 " " 5 2 Dz. Chickens * 1.37 2 7.56 1 9. " 123 2 Ibs. Butter 18 3 23.15* Cr. "$225^97 Ap. 4. By 25 Hogs, 3147 Ibs 3 2 110.14 2 9. 1 Plow 7.62 2 " Cash 108.20 $225.97 JONATHAN FARMER. MECHANIC'S BILL. ST. Louis, July 3, 1867. MESS. BUREAU & Co., To JOHN WORKMAN. Jan. 1. For Cab't W'k on 3 dz. hf-Fr. Bed's. 20.00 60.00 Feb. 9. " " " " 4" 2-pn'ldo... 30.00 120.00 Cr. 180.00 Jan. 1. By Cash on Ace 10.00 " 8. " .Order on W. Shoemaker f 15.00 " 15. " Cash 20.00 45.00 Settled by note at 90 ds $135.00 JOHN WORKMAN. When the signature of the merchant is required to a part pay- ment, the phrase Balance due should be written opposite the amount unpaid. * Express the 137 2 in mills before multiplying. t For form of order and note see index. Hf. Fr. Bd's.> half French bedsteads; pnL, panel. \ REDUCTION. 117 DRESS-MAKER'S BILL. BOSTON, Sep. 9, 1866. MRS. AFFLUENT, To M. E. FASHION. For Making 1 Moire Antique Dress 25.00 " Trimming do 75.00 $100.00 XII. COMPOUND NUMBERS. BESIDES the distinction made between numbers in chap- ter viii, they may be divided into simple, and compound. 108. A simple number may be abstract or concrete, of one denomination, as 27 men, 35 dollars. 109. A compound number is always concrete and com- posed of more than one denomination, as 157 dollars 50 cents, 29 pounds 14 shillings and 6 pence two numbers, each expressing one sum of money. 110. Reduction of compound numbers. 111. REDUCTION is the process of changing concrete numbers of one denomination to those of equal value in another. If I multiply 5 bushels by the number of pecks in a bushel, I reduce them to pecks, and thus change both the number and denomination, while I preserve the value.* 112. Changing the denomination from a higher to a lower, as bushels to pecks, is called reduction descending; while the reverse process, as changing pecks to bushels, is called reduction ascending. A few examples will suffice to teach all that is neces- sary to be known on this subject. * On page 70 are several exercises which properly belong to this subject. 118 NELSON'S COMMON-SCHOOL ARITHMETIC. 1. To reduce 35 acres to, or represent them, in square feet. 35 Acres EXPLANATION. In 1 acre there are 4 roods, 4 in 35 acres there are 35 times 4 roods (35X4) TT7 r> 7 or 140 roods; in 1 rood there are 40 square rods, in 140 roods there are 140X40 or 5600 sq. rods ; in 1 sq. rod there are 30J sq. yards, 5600 Sq. Rods j n 5^00 S q. rods there are 5600X30J or 169400 sq. yards ; in 1 sq. yard there are 9 sq. feet, 30j 168000 in 169400 sq. /ards there are 169400X9 or 1400 1524600 sq. feet. REMARK. It will be observed that the game regulfc would haye been produced by multiplying the 35 by 43560, the number of 1524600 feet in an acre. When there are items between the highest and the low- est denominations, they should be added, as shown in the following example: 2. In 75, 13s, 6Jd how many farthings? 75 13 6J 20 1500 Shillings in 75. 13 Shillings added. 1513 Whole number of shillings. 12 18156 Pence in 1513 shillings. 6 Pence added. 18162 Sum of pence. 4 Par things in 1 penny. 72648 9. 72650 Parthings in 18162J pence. To add the items mentally, the 3 shillings would occupy the place of the first 0; then, multiplying the 5 by the 2, we would obtain 10 to which might be added the 1, making 11; then, multiplying the 7 by 2, we get 14 and the one carried, making 15 or 1513 at once. REDUCTION. H9 3. In 3 miles 21 rods how many yards? 4. In 145 tons 25 Ibs. of hemp how many pounds? 5. How many pence in 197, 17s, 9d? 6. How many farthings in 57, 13s, 6? 7. In 93 barrels of apples how many pecks, each barrel containing 2 bushels 3 pecks? Answers: 324825, 55370, 47493, 1023, 5395. 113. To reduce concrete numbers of a lower denomina- tion to those of a higher, the process will be the reverse of the last. 8. Reduce 72650 farthings to pounds. 4)72650 Farthings. In farthings there will be one fourth 1 9M m t\9 9 P as man y P en ce; in pence one twelfth as ~ 1 ' many shillings; and in shillings one 2 1 0)151 1 3 6 Skill. twentieth as many pounds. The re- 75 13 6 1 raainders are 13 shillings, 6 pence and 2 farthings or J penny. 9. Reduce 4163 linear inches to yards. 10. Express 31456739 minutes in years, months and days, allowing 365 days 6 hours to the year.* 11. Reduce 456372 farthings to pounds. 12. At 1 mile in 4J minutes, how many miles would a locomotive run in 5 hours? 13. A ship sailed 3000 miles in 16 days; what was her average speed per hour? 14. In 35 cubic yards how many cubic inches ? 15. In 5J square rods how many square feet? 16. Divide a log 55 feet in length into 15 equal -parts, and express the result in feet and inches. * Reduce the year to minutes and divide them into the minutes in question, which will give the number of years. The remainder being minutes, may be reduced to hours, etc., as in reduction. 120 NELSON'S COMMON-SCHOOL ARITHMETIC. 17. What will be the cost of 35f bushels of strawber- ries at 15 cents a quart? 18. At 5 cents for 3 sheets of paper, how much money can be obtained for 35| reams, allowing half price for the outside half quires (35-J- qrs.) of each ream? 19. A merchant buys 15 barrels of potatoes, containing 2 bushels 2f pecks each, at $1.25 per bushel, and sells them at 25 cents a half peck; how much does he make, allowing \ a peck per barrel for loss in measuring? 20. In 316754 Ibs. of hemp, how many tons, cwt., etc.? Answers: 141, 8, 18; 17160; 3, 8; 66| ; 945; 27690; 59, 9, 25, 4, 59; 49|; 475, 7, 9; 7jf ; 26.48 T \; 1497|; 1632960. 114. To add compound numbers. 1. What is the amount of the following sums of British money ? s. d. SOLUTION. 1. We first add the fractions, calling 18 17 4^r them farthings, which makes 6 farthings; these we 19 g 7JL reduce to pence by dividing them by 4. f If o p 17 7 83- *-' Wr * te ? ancl a(1( * tlie * P ennv to tn e column of 5 pence, which makes 20 pence; this number divided 55 11 8-J- by 12 (the number of pence in a shilling)^! shilling and 8 pence. Write the 8 under the pence, and add 1 to the units of the shillings' place, which makes 21; write 1 and add the 2 to the ten's column=3 or 31 shillings, which, divided by 20 1 and 11 shillings left. Write the latter under the shillings and add tho 1 pound to the pound's column=r55. Ans. <55, 11s, 8J. Add the following: 2. 17 18 llf + 14 17 2J+ 16 14 8 =? 3. 17 19 OJ+ 45 11|+111 10 2J=:? t. 116 16 6 +320 14 5{-+ 38 18 8 =? 5. 24 18 6 + 180 10 Of+ 66 19 11|=? 6. 175 19 7|+ 90 8 8|+575 12 6=? 7. 201 17 6|+1010 10 10]+970 19 11|=? Totals, 3297 17 9f 700, 10s, 8d. Hhtftk^- COMPOUND NUMBERS. 121 115. To subtract compound numbers. 1. From 19, 4s, 4d take 14, 7s, 6]d. s. d. EXPLANATION. We can not take J from nothing, 19 4 3 so we add a penny to both terms; subtracting -J- from 14 7 6^ the 1 penny, or 4 fourths, we have Jleft. Adding Id ^T"T^ ^ to the 6d we have 7d, which we can not subtract * from the 3d above, and accordingly add Is to both numbers. 7 from Is 3d or lod, leaves 8d. Adding Is to the shil- lings, we have 8s, which can not be taken from 4s without adding 1 to both numbers; 1 to 4s=24s; 8s from 24s=16s. Then adding 1 to the 14, we have 15, which, taken from 19=4, making the answer 4, 16s. 8Jd. Subtract the following: s. d. s. d. 2. 17 10 8i 14 53:=:? 3. 119 7 6 17 19 5J=? 4. 500 20 18 8 =? 5. 176 14 7] 129 15 7%=? Total, 630, 13s, 9Jd 116. To multiply compound numbers.* 6. Multiply 17, 4s, 9J by 8. OPERATION. 17 4 9| 137 18 2 After performing operations in addition, the learner will readily see how this is done. 7. 17 18 8J X 7==? 10. 48 9 6JX2 and 3=? 8. 120 16 61X12=? 11. 145 8 9. 365 7|X 9=? 12. 705 13 Total, 4860, 15s, OJd. Total, 5639, 19, 6. * Multiplication may likewise be performed by reducing the com- pound number to one denomination. (See Reduction) 11 122 NELSON'S COMMON-SCHOOL ARITHMETIC. 117. When the multiplier exceeds 12 and is a composite number, or otherwise. 13. 48, 9, s. d. EXPLANATION. Here we multiply by the two 48 9 6J factors of which the 24 is composed. 3 145 8 7^=3 times the amount. 1163 9 =8 times 3 or 24 times the amount. 14. Multiply 705, 13s, 9|d by 38. 705 13 9|X2 4234 2 10|=6 times the amount. 6 25404 17 3 =6 times 6 or 36 times the amount. 1411 7 7^=2 times the amount. 26816 4 lQ$=Sum 0/2+36 times or 3%Xthe amount. 15. 19 6s 7 dX 84=? 18. 27 8s 8 dX 87=? 16. 91 18s 5idX 89=? 19. 77 17s 7|dX 95=? 17. 4s 7JdXl29=? . 20. 89 17s 6 dx!50=? 21. 176 Xl7=? 24. 5 7 6|X26=? 22. 349 X19=? 25. 638 X29=? 23. 4 5 7^X23=? 26. 8 4 7|X30=? Answers: 23266, 15, 4|; 9834, 12, 5; 566, 1, 3; 183, 7, 11. 118. To divide compound numbers. British money being almost the only thing in business to which compound numbers is applied exercises in it have received most at- tention; and especially as direct importation gives the clerk more to do with it than heretofore. See index for shorter methods of computing this kind of money COMPOUND NUMBERS. 123 27. Divide 157, 13, 6J, by 5. EXPLANATION. 5 is contained in 157 31 s. d. times and 2 over. These 2 reduced to shil- 5)157 13 6J- lings, and added to the 13s. of the dividend, 31 10 8^- make 53s., in which 5 is contained 10 times and 3s. left. In 3s. there are 36 pence, which, added to the 6d. of the dividend, make 42d., in which 5 is contained 8 times and 2d. over. In 2d. and |d. there are 10 farthings, in which 5 is contained 2 times, making j or ^d. 28. Divide 157, 13s, 6Jd equally between 25 persons, 25) 157, 13s, 6Jd(6, 6s, IJd, or 6, 6s, l|d, nearly. 150 7 =remainder. 20 W3=shilUngs in 7, with 13s of the dividend added. JL50 3=remainder in shillings. 12 42=pence in 3 shillings and 6 pence from the div d. Yl=remainder in pence. 4 70=/ar things in 17 pence and \. 50 20=remainder) or |-5=-f farthings. s. d. s. d. 29. 487 13 -r- 9=? 32. 167 18 6|-5- 25=? 30. 356 7 10 -r-36=? ' 33. 768 14 3 J -5-125=? *31. 419 15 6-r-14=? 34. 17 11 3|-r-875=? Answers : 12, 17, 8|, and 94, 1, 4|. * When the remainder from farthings is J or over, add a farthing, otherwise omit it* 124 NELSON'S COMMON-SCHOOL ARITHMETIC. GENERAL EXERCISES IN COMPOUND NUMBERS. 28. In 4 yards 2 feet seven inches, linear measure, how many inches? 29. In 100 inches how many yards? 30. Reduce 3520 yards to miles. 31. In 100,000 inches how many miles? 32. How many times will a carriage-wheel turn in a distance of 17 miles, the wheel measuring 2 yards 2 feet in circumference? 33. From a plank 17 yards long was cut 10 yards 2 feet 3 inches; how much of it was left? 34. Reduce 3 acres 140 rods to square yards. 35. Divide 200 acres 100 rods into 10 equal parts. 36. From 406 acres 17 rods take 68 acres 148 rods 15 yards. 37. Divide 64 acres 134 rods 8 yards into 5 equal parts. 38. Find the price of 9| ounces of gold at 3 17s 8d per Ib. Troy.* 39. Find how often 2 4s 6d is contained in 41 10s 7>d. 40. Reduce 1 Ib. 3 oz. 5 pwt. to pennyweights. Answers: 175; 11220; 1, 4, 24, 5, 2, 4; 2; 2, 2, 4; 6, 9; 337, 28, 15} ; 2, 19, 10^; 12, 3, 34, 25; 18|||; 18755; 20, 10 ; 305. The Teacher should require his scholars to give the denomination of each item in the answers. *Find the price of 1 ounce j then the price of 9J. SHORT METHODS OF MULTIPLYING. 125 XIII. SHORT METHODS OF MULTIPLYING. 119. BESIDES the contractions by aliquots, under Art. 66, the expert accountant and arithmetician can find ab- breviated methods adapted to almost every calculation. A few will be given in this place, to admit of the learner using them, when opportunity OCCIMTS, in the subsequent exercises. 120. To multiply by 11, write the first figure of the multiplicand as the first of the product, and add each figure on the left to the one on the right, as below. 1. 38397X11=427867. Prove the following by multiplying in the ordinary way : 2. 379X11=? 7. $219.168X11=? 3. 1487XH=? 8. $716.573X11=? 4. $37.486X11=? 9. $316.144x11=? 5. $9314.20 XH=? 0. $137.211X11=? 6. $167.473X11=? 11. $710.22 XH=? 121. To multiply by the teens when the tables are not known, and by such numbers of two figures as end with 1, as 21, 31, etc., multiplication by the figure 1 ought to be omitted. 12. Multiply 3174 by 17. 3174 EXPLANATION. The product of 7 is written one place 22218 to the right to allow the first line to stand in the tens' place, by which it is multiplied by 10. 13. $3163 X15= ? 18. $435.16fXlG=? 14. $216.37 X19=? 19. $213.14 Xl8=? 15. $1139.24 X13=? 20. $1137.37^X19=? 16. $413.22 X18=? 21. $713.1Hxlti=? 17. *8131.18|X14=? 22 - $4302.87 X 17=? *Call the J of a cent 25 hundredths, making 1311825 for the multiplicand, and point oft' four figures. (See note, page 06.) 126 NELSON'S COMMON-SCHOOL ARITHMETIC. 23. $316.27X51=? 31627 EXPLANATION. The 5 of the multiplier being in the 158135 tens' place, the first figure of the product -is written if 19 077 under the tens of the multiplicand. 24. $137.50 X31=? 25. $298.67 X&1=? 26. $783.37JX61=? 27. $313.17 X81=? 28. $1987.871X91=? 29. $2136.22 X 71=? 30. $1394.311 X 41=? 31. $653.18|X 21=? 32. $291.16* Xl21=? 33. $312.18|X 21=? 122. When the multiplier wants from 1 to 12 of being 100, 200, 3000, etc., the work may be contracted by mul- tiplying by one of these, and subtracting as many times the multiplicand as the multiplier is short of it. 34. To multiply 424 by 97. OPERATION. 424x100=42400 424X 3= 1272 35. 765X192=? 30. 1789X398=? 37. 67.84X188=? 9876X191=-? 671 X B9=? 59X689=? 38. 39. 40. 41128 $89 $167 $37.98 $478.96 X784=? X 29=? X489=? X499=? 41. 42. 43. 44. 45. $674.82^X992=?' 46. $7164.37-^X 87=? 123. When the multiplier is 29, 39, 49, etc., we multi- ply by the next higher number and subtract the multipli- cand, 47. To multiply 176 by 59. OPERATION. 176X60=10560 J 176 . 10384 124. To multiply ly 601, 1003, 90001, etc. The case differs from Art. 8 only in the intervening SHORT METHODS OF MULTIPLYING 127 figures; so the product is written one place further to the right or left for every cipher. ^48. Multiply 317 by 601. OPERATION. 317 1902 Ans. 190517 49. Multiply 15704 by 10007 OPERATION. 15704 109928 Ans. 157149928 125. When one part of the multiplier contains the other without a remainder, as 248. Here 24 contains 3 times the 8 or first figure, so by multiplying the product of 8 times the multiplicand by 3, one line is saved. 50. Multiply 76439 by 248. OPERATION. 76439 611512=8 times 76439 1834536_=3 " 611512 18956872 Ans. REMARK. This operation might be shortened by multiplying the product by 8 mentally, and adding that line for the whole product. 51. Multiply 25938 by 936. OPERATION. 25938x936 or _ 25938 233442 ~~ 233442.. 933768 24277968 24277968 52. 11457X324=? 57. 7832x64256=? 53. 672x189=? 58. 7498x16144=? 54. 783x357=? 59. 9739X 3972=? 55. 924X218=? 60. 6487X 8109=? 56. 596X426=? 61. 74675X 7206=? 126. To multiply l>y 375, 625, 750 or 875, we first mul- tiply by 125 (Art. 66), and that product by 3, 5, 6 or 7, these numbers, 375, etc., being multiples of that number. 128 NELSON'S COMMON-SCHOOL ARITHMETIC. 1649000X5 8245000 62. 1649X625= g^-= g 1030625 63. 3156X375=? $1703.20x750=? $1456x875=? 127. To multiply by a composite number composed of factors under 13, the latter may be used as multipliers instead of the former. 64. 314X72. OPERATION. 314x12=3768x6=22608. 65. 932X64=? 68. $913.27 X^4=? 66. 738X48=? 69. $293.75 X72=? 67. 426X96=? 70. $6318.371x63=? 128. To multiply by any number of 9s, we multiply by the next highest number and subtract the multiplicand. 71. 3145X999- 3145X1000=3145000 Prom which take 3145 3141855 129. To square, mentally, numbers under 39 that end with 9. 72. What is the square of 29? 29 EXPLANATION. Writing 1 for the first figure of the pro- 29 duct, we add 1 to the tens' place of the multiplier, and rnul- 7777 tiply the sum on the multiplicand less 1 : 3X28=84, with the 1 annexed=841. 73. Find the square of the following numbers mentally: 99, 59, 119, 79, 19, 69, 129, 89. 130. To square any number of 9s instantaneously, and without multiplying. Commencing at the left, we write as many 9s, less one, as the number to be squared, an 8, as many Os as 9s and a 1. 74. The square of 9999999 is 99999980000001. SHORT METHODS OF DIVIDING. 129 The square of any number of 3s will be one-ninth of the square of the 9s. 131. To square numbers under 135 ending with 5. The first two figures on the right of the product will always be 25 ; and to find the others, we add 1 to the tens' place and multiply it on the tens' and hundreds' places above. 75. To square 115. OPERATION. 11 12 13225 The reason of this will be apparent by multiplying in the usual way. 132. To square a number containing a half, as 12J, we multiply the whole number by the next higher number and add a fourth. 8| squared=8x9-|-{=72J. 76. Find the square of the following numbers: 99999, 33333, 75, 45, 65, 62 16J, 19J. XIV. SHORT METHODS OF DIVIDING* 133. Division may often be contracted by cancellation* tvhen the terms are written in fractional form. 1. Divide 1463 by 28. 209 EXPLANATION. The terms 1463 and 28 were first divided by 7, leaving 209 fourths, and 209 divided by * 4 S ives 52 i- To THE TEACHER. The author does not offer all these contrac- tions as rules of general utility; still, he is of opinion that a fa- miliar knowledge of them will be advantageous to the student of arithmetic in disciplining his mind and showing him the relation of numbers. Where the instructor thinks otherwise, he can omit them. * To cancel signifies to blot out or make void. 130 NELSON'S COMMON-SCHOOL ARITHMETIC. Prove the answers obtained, by dividing in the usual way. 2. 3465-v-35=? 2763-4- 81? $65.45--243= ? 3. 1962-r-22=? 6876-r-152=? $54.36-r-144=z? 134. To divide by aliquots of 100, 1000, etc. This process is the reverse of that under Art. 66. 135. To divide by a composite number, as 96, which is composed of the factors 12 and 8, or 648, which is com- posed of 9X8X9. This operation is performed by using the factors instead of the whole number. 4. Divide $78.54 by 32. OPERATION. 4)7854 8)19632 2453 How the true remainder is found: The first remainder is 2 cents, because it is left from the cents that were divided. The second remainder is four times as great as if it were from the first line, because every figure of the second line is four times as great as if it stood in the first line. Four times 312 and the 2 of the first remainder equal 14, the true remainder. Ans. 245f . 5. Divide 6371 by 336. OPERATION. 6)6371 7)10615 REMARK. The true remainder of this example is - ~~ found by multiplying the last remainder by 7, to '_l__ make it of the same value as if it were from the line 18 7 above, and that by 6, to make it of the same value as if it were from the upper line: 7X?X G:=r: 2H to which add 6X4+5 or 29. The true remainder is 323. Ans. 1 8|||. 6. Divide 1463 by 28 10. 4571-f-441==? 7. " 7614 " 72 11. 1987-^379=? 8. " 1943 " 49 12. 9843-^-720==? 9. " 8765 " 343 13. 1456-f-729=? SHORT METHODS OF DIVIDING. 131 14. Divide 7654 by 25. 76.54=10<M of 7654. 4 306.16=4 times as much, or ^ of 7654. or 306^=306^. 15. $3675-^-125 = ? S213.67-s-16=? $11 74-4-121=? 16. $2153-- 33$=? $319.25--- 8$=? 316-^25 =? 136. When the divisor is 15, 35, 45, 55 or 65, it will abbreviate the work to multiply the dividend by 2 and di- vide by 30, 70, etc. 17. 345-^-35=? 345x2=690-^-7=98f 18. 2756-4-15=? $1324.25-4-35? $365.75-4-45=? 137. To divide by 75, 175, 225 or 275, the dividend may be multiplied by 4 and the product divided by 300, 700, 900 or 1100. 19. 2136-4-75=? 2136 EXPLANATION. 4 three-hundred ths being equal 4 to 1 seventy-fifth, it abbreviates the work to divide 3 00^)85 44 b ^ ^^ an( * mult *P 1 y by 4. 20. 3678-4-175=? 6317-r-175=? $19.32-f-275=? 138. Long division may be abbreviated by performing a part of the process mentally, and writing only the result. 21. 76354-^-34=? 34)76354(2245 EXPLANATION. The products are omitted; only 83 the remainder and the figures brought down are 155 written. 194 24 22. 3167^-184=? 1679-i-21=? 67831-4-498=? 139. To divide by any number of 9s, the dividend may be pointed off, from the left, in periods corresponding with 132 NELSON'S COMMON-SCHOOL ARITHMETIC. the number of figures in the divisor. If the divisor have two 9s, it may be pointed off in periods of two; if three 9s, in periods of three, and the operation performed as in the formula. 23. 316328-;-99^? 7312961--999=? 31,63,28 731,296,1 REMARK. It will be observed that 31,63 731,2 each line in the formula is diminished 31 7 by one period, corresponding with the 01 OR oo ^7OOA OQA number of 9s, and the sum of the whole oiyo.-u.iu lo^u.^oU , . ., p. , , . 1 -. taken. As many figures as Us should I ~ be pointed off for a remainder, plus 3195.23 7320.281 the carrying figure of the first line. 24. Divide 167389 by 9999 and 7654321 by 99.* XV. MARKING GOODS GAIN AND LOSS. 140. ON receiving goods for sale, the merchant or some of his clerks 'examine them by the bill or invoice, which is usually received in advance of them by mail, after which they proceed to mark them. 141. Marking Goods. This is done by selecting sam- ples of each kind or quality of goods, and putting on them a private mark, indicating the cost price, the selling price, or both. Every house has its own private mark, which usually consists of a word or phrase to represent the ten digits, as the word importance, which has ten letters. Commenc- ing at 1, the letters are arranged as follows: importance 12 34567890 In addition to these, another letter not contained in the above is selected for a repeater, so that when a figure oc- * It would be no abbreviation to divide by 9 in this way. MARKING GOODS GAIN AND LOSS. 133 curs twice it may be inserted to prevent detection. The letter g will do in this case. To mark *6.55, the letters trg would be used, with per- haps a line or period to separate the dollars from the cents. To fix on a selling price, various circumstances have to be taken into consideration-^-the cost of transportation, the probable length of time required to sell, cost of rent, wages of clerks, depreciation of stock, etc. A certain rate is put on for the first item, say 5 to 15 per cent.; then an amount estimated to cover the balance and leave a profit. When adding the entire rate of ad- vance to the cost price, the clerk is not required to be exact, as simplicity of calculation is a greater object than uniformity of profit. Thus, in marking goods to sell by the dozen, a multiple of 12 would be preferred, whether a little above or below the fixed rate. For the same reason, aliquots, or numbers formed of aliquots, of 100 would be selected. When working the following exercises, the learner will remember that fractions, unless occurring with aliquots of 100, are usually omitted. When under ^ they are rejected; otherwise, a cent is added to the cents/* 1. Add 25% to $3.50, $5.75, $1.82, .75, $2. Answers: $4.37 2 , $7.18f, $2.28, .94, $2.50. 2. Add 16|- % to $6.20, $3.12 2 , 27c, $3.87. 3. Add 37% to 20c, 12 2 c, $1.15, 6c, 15c. 4. Add 15% to $1.20, 75c, $1.22, $3.57, 16 2 c, 27c. 5. Add 33J% to 5c, 15c, 21c, $6.75, 18c, $27.50. Answers: $15.72, $8.24, $46.46, $2.32. *To indicate the degree of exactness required, a few answers will be given to the examples which follow. One object in giving these examples being to exercise the judg- ment of the pupil, he will not be required to obtain precisely the same figures as given in the other answers. 134 NELSON'S COMMON-SCHOOL ARITHMETIC. 6. $1.75+14% = ? 14. $6.75+26%=? 7. .87+12% = ? 15. .07+40% = ? 8. $1.67+53% = ? 16. $3.60+12% = ? 9. .04+60% = ,? 17. $4.80+30% = ? 10. $2.75+35% = ? 18. $1.20+35% = ? 11. .16+15% = ? 19. $7.20+40% = ? 12. .05+25%=? 20. $6.00+14% = ? 13. $3.14+18%=? 21. $5.10+25% = ? Answers: $13.25, $43.79. 22. To .30 add 20 % profit. 23. To $1.20 add 5 % charges and 20% profit. 24. To $1.75 add 2J% freight and 10% profit. 25. To .08 add 16| % profit. 26. To .36 add 33J % profit. Answer: $4.41. 27. $0.10+10%=? 30. $1.00+16|%=? 28. .05+50%=? 31. $3.25+334% = ? 29. .25+20%=? 32. $1.40+12^%=? Total, $7.57. The following may be worked by aliquots of 100: 33. Add 5 % to $12.50 44. To $3.50 add 4 % 34. " 6|-% " $7.80 45. " $9.50 " 35. " 84% $4.85 46. $0.10 " 36. " 9 % " $3.50 47. " $0.18f " 37. " 124% " $0.87 48. " $0.06^ " 40 % 38. " 20 % " $0.45 49. " $0.87^- " 50 % 39. " 18f% $0.15 50. " $0.084 75 % 40. " 25 % $0.124 51. " $0.11J" 20 % 41. " 7 % " $6.20 52. " $0.80 " 42. 34% " $1.00 53. " $0.16 " 43. " 12J>% " $1'.35 54. $0.47 Answers: 41.53, $18.56. GOODS GAIN AND LOSS. 135 ORAL EXERCISES .* 1. $0.10+ 25 % = ? 18. $0.50 + 20 % = ? 2. .25+ 10 % = ? 19. .25 + 50 % = ? 3. .20+ 6 % = ? 20. .11 + 15 % = ? 4. .33+ 33J% = ? 21. .12^+ 20 % = ? 5. .16+ 25 % = ? 22. .45 + 9 % = ? 6. .45+ 5 % = ? 23. .66 + 33|% = ? 7. .87+ 30 % = ? 24. .14 + 5 % = ? 8. .16+ m%=? 25. .35 + 20 % = ? 9. .10+ 22 %=*\ 26. .16 + 50 % = ? 10. .05+ 30 % = ? 27. .67 + 12 % = ? 11. $2.20+ 50 % = ? 28. $1.20 + 16|% = ? 12. $550 + 100 %=? 29. .27 + 50 % = ? 13. .75+ 60 % = ? 30. $6.50 + 16 % = ? 14. $1.25+ 30 % = ? 31. $2.15 + 8 % = ? 15. $1.12+ 25 % = ? 32. $1.87 +200 % = ? 16. .75+ 20 *T=? 33. .31 + 18 % = ? 17. 1.50+ 8 % = ? 34. .19 + 12% = ? 35. 30% of $20.00=? 43. $0.37^+30 % = ? 36. 50% of $1000.00==? 44. .62J+ 5 % = ? 37. 25% of $700.00 = ? 45. $1.25 +12J% = ? 38. 16% of $500.00=? 46. .16 + 5 % = ? 39. 10% of $350.25 = ? 47. .54 +40 % = ? 40. 15% of $200.00=? 48. $30.00 +60 % = ? 41. 80% of $500.00=? 49. $90.00 + 9 % = ? 42. 20% of $20.00=? 50. $70.00 +30 % = ? * The Teacher may require the flections ill these exercises. 136 NELSON'S COMMON-SCHOOL ARITHMETIC. GAIN AND LOSS. 142. Merchants distinguish between real gain or loss and gain or loss per cent., calling the former the actual gain or loss, and the latter the gain or loss per cent. 143. To find the actual gain, it is simply necessary to subtract the cost price from the selling price. 1. Bought a house and lot for $4367 and sold them for $5000; how much did I gain? =(7os or buying price. $633=:-4c2waZ gain. Cost price. Selling price. Cost price. Selling price. 2. $2.75 $4.87 7. $316.17 $215.25 3. $97.35 $120.10 8. $112.14 $120.48 4. $6.87 $6.98 9. $317.18| $21.9.m 5. $5.40 $9.80 10. $67.21 *8&2J8 6. $3.20 $6.40 11. $54.12J 821.18J Total gain, $32.58. Net loss, or loss with gains de- ducted, $204.51. 144. To find the gain per cent., is simply to find the gain on every hundred dollars or cents. Required the gain per cent, on goods which sold at $1.35 and cost $1.20. 135120=15 cents, the actual gain on 120. 15-=- 120= T yg=gain on 1 cent. y'&X 100=^^=12 J per cent., or gain, on 100 cen-ts. OPERATION. 135 EXPLANATION. The actual gain 120 is first found; then the gain per cent., by dividing the actual gain ( when multiplied by 100) by ti& first cost. MARKING GOODS GAIN AND LOSS. 137 12. Goods which cost $2.00 were sold at $3.00; required the gain per cent. 13. The cost price was $1.25; the selling price, $1.50; what was the gain per cent.? 14. Goods bought at 75 cents sold at $1.00; what was the gain per cent. ? 15. 10 cents was the cost; 12 \ the selling price; re- quired the gain per cent. Total rates of gain of the four, First cost. Selling price. First cost. Soiling price. 16. $12.50 $10.00 20. $3167.00 $3000.00 17. .18 .20 21. $1000.00 $1500.00 18. .05 .06 22. $27.80 $20.00 19. $127.52 $111.58 23. $12.17 $11.50 Net loss, on 16 to 19, \-^%. Answers to second group, 24. Bought a bbl. of apples for $1.75 and sold it for $2.25; what did I gain per cent.? 25. Sold 25 bbls. of potatoes for $39.00; how much did I gain per cent., if they cost me $1.25 per barrel? 26. Bought 150 bbls. of flour @ $5.25, paid for dray- age $7.50 and porterage $1.00; at what per barrel should I sell it to gain 15 per cent.? 27. Bought 15 horses at $125 each, and sold the lot for $3500; what was my gain per cent., after paying $25 for their feed? 28. Sold a safe which cost me $80 for $75; what was my loss per cent.? 29. Bought a bill of goods for $350, paid freight $15.20, insurance $5, drayage $3, and sold them for $425; what was my actual gain, and what my gain p^er cent.? 30. Sold A's note for $750 at a discount of 15%; what did I pay for it? 12 138 NELSON'S COMMON-SCHOOL ARITHMETIC. 31. Sold B's note of $320 for $300; what was the rate of discount? Answers: $637.50, $51.80, 14% nearly, 84J% nearly, 6 i%> 6 f%> 28 T%> 24 f%> $ 6 - 10 - 145. When the selling price and the rate per cent, are known, to find the first cost. 32. What was the first cost of goods marked $2.65, the rate per cent, of profit being 25? 1.25)2.65(212 EXPLANATION. Every dollar invested in 2 50 or $2.12 the goods has increased 25 cents, and is - worth 125 cents. Hence, there are as many irQfl invested dollars in $2.65 as 1.25 is con- tained times in it. The remainder, 15 dol- lars, we reduced to cents, which, divided by 1.25, gives 12 cents. Another way: Since 25% is J of 100, the $2.65 must be I more than the first cost. Let the first cost be f, then J+|=|. Therefore, $2.65=f of the first cost. 2|5. = 53=one-fourth of the first cost. 53X4=2.12, first cost. What was the first cost of the goods marked as follows? The learner can prove his calculations by reversing the process. 33. $2.25 @ 10 % gain 38. $2.87 @ 10 % loss 34. $3.70 " 5 % " 39. $1.54 " 6 % " 35. $115.87 " 12%% " 40. $3.75 " 25 % " 36. $14.54 " ^1% " 41. .87J " 12%% 37. .87 " 16f% " 42. .12% " 5o"'% " 43. $9.50 @ 50 % gain 47. $90.00 @ 20 % gain 44. $7.87 " 25 % " 48. $75.30 " 15 % 45. $6.50 " 16f% " 49. $82.50 ' 46. $8.75 18f % " 50. $60.00 20 COMMISSION AND BROKERAGE. 139 XVI. COMMISSION AKD BROKERAGE* 146. COMMISSION, or brokerage, is the percentage charged by a commission merchant, factor, agent or broker for transacting business for another. 147. Commission is usually reckoned on the whole amount of sale, purchase or collection. 1. At 2J per cent., what is the commission on $17640? Ans. $441. 2. A merchant sells goods for another to the amount of $4371.81; what is his commission at 5 per cent.? 3. A broker receives | per cent, for selling $2500 worth of merchandise for a commission merchant; what is the amount of his brokerage? 4. A of New Orleans buys sugar for B of Cincinnati to the amount of $7100; what is the amount of his commis- sion at 1J per cent.? 5. A commission merchant sells goods for his principal to the amount of $3000, and charges 2^ per cent, commis- sion; what does he make by the operation, after paying a broker | per cent, for his services in effecting sales? 6. After receiving 5 per cent, commission on sales amounting to $520.75, how much should I return to my principal? 7. Gave a lawyer a note of $50 to collect, at 8 per cent. ; how much should I receive? Omit fractions of a cent in the answer: Answers: $106.50, $6.25, $218.59, $67.50, $494.71, $46, $510.71. ; *See Commission Merchant, page 94; Brokers, page 95. 140 NELSON'S COMMON-SCHOOL ARITHMETIC. What is the commission on the following amounts? 8. $364.15 @ 3$i = ? 12. 36.21 @ 9. $78.54 6^% = ? 13. $174.09 10. $710.06 83% = ? 14. $2167.90 11. $876.75 " 9J%r=? 15. $78.21 " $% = 1 Total,*$161.31. Total $146.13. 16. A commission merchant charges 2J% com. and 2^r% guarantee on the sale of goods amounting to $3100; how much should he return to his principal? 17. A merchant receives a consignment of goods valued at $3000, and sells them for $4500 ; how much should he remit to the consignor, after reserving 4J% for com. and guar. ? Answers: $2945, $4297.50, $5297.30. 18. A merchant sells a note of $100 to a money broker, at a discount of 6 per cent.; how much money does he receive? 19. A New York merchant buys a bill of exchange worth $400, on a Cincinnati banking house, at J per cent, discount; how much does he pay for it? 20. Bought a bill of exchange on New York for $7691, at l^r premium; what did I pay for it? 21. Purchased 20 shares of railroad stock, worth $20 per share, @ 12^% discount, and sold it at par; what was the amount paid? and how much did I gain? Answers: $94, $399, $7806.37, $350, $50. 22. "What is the premium on the following? $31.46@ 5J%; $1760@6i%; $4617@9i%. Total, $550.35. 148. To find tlie commission on investments. The merchant often has moneys in his hands or re* mitted to him for investment in goods or stocks, upon which he is allowed commission on the amount invested only. COMMISSION AND BROKERAGE. HI 23. At 2-|-% commission, what amount of money shall I retain of $2000 in my hands for investment? This $2000 contains 100% of the amount to be invested, plus 2.5%, my commission making 102,5% of it. Reducing both to tenths, we have 2000.0 to be divided by 102.5. 102.5)2000.0(19.5121 or $19.5122=1%, 1025 Which, multiplied by 2J gives the com- 9750 mission, or by 100, gives 100 per cent., or 0995 the amount to be invested. Com $48.78. 5250000 REMARK. The 525 dollars remainder were 5125 reduced to tenths of mills. 125<T 1025 2250 2050 2000 1025 975 RECAPITULATION. To find 1 per cent., we divide 2000 by 102.5, which, multiplied by 100, gives 100 rer cent., or the amount. The difference is my commission. 24. What will be my com. on $1300, to be invested at 2%? 25. At 5% commission, how much sugar can I buy for $3475.25, when it sells at 20 cents a pound? 26. Out of $987.50 how many pounds of tobacco should I purchase for my principal at 35 cents a pound, commis- sion 2|%? 27. At 2|-% commission, how much shall I reserve of $2130.67, after investing in cotton at 25 cents a pound? 28. To pay 12-J- cents a pound for rice, and -fa% for insurance, and myself 3% commission, how many pounds can I buy for $345.15? The learner will prove his work by reversing the process. 142 NELSON'S COMMON-SCHOOL ARITHMETIC. 29. Invest $1367.37, \ in gloves at 30 a dozen, \ in sheeting at 35 cents a yard and the balance in muslin at 20 cents a yard; commission 2J%. INTEREST. 149. Interest is a percentage allowed for the use of capital, which may consist of money, merchandise or debts due. It is regulated by the year or month. 150. The sum upon which interest is reckoned is called the principal; the percentage allowed per month or year, the rate; and the sum of the principal and interest, the amount. 151. Interest is divided into simple and compound Simple interest is percentage on the principal alone; com- pound interest is interest reckoned on both principal and interest. The legal rates of interest in the several States is vari- able. 152. Money loaned on interest is usually secured by promissory notes, expressed as follows: 153. This note would be due on the 1st day of Decem- ber but for a law which allows three days longer for its payment, called days of grace. It will therefore be legally due on December 4th, and interest will be reckoned to that date. * Other forms of notes will be found under Banking. SIMPLE INTEREST, 143 XVII. SIMPLE INTEREST. 154. Interest is usually calculated on the basis of 360 days to the year.* When notes are drawn by the month, calendar months are understood. Thus, a note drawn on the 1st of September, as the preceding, falls due on the 1st of December, plus the days of grace. 155. The simplest method of computing interest is to do it at the rate of 6 per cent, per annum, and add or subtract when it is higher or lower than that. 156. The interest for 60 days at 6% per annum is equal to as many cents as there are dollars, or, in other words, is 1 per cent, of the principal. The reason of this is obvious. 6 per cent, per annum is ^ per cent, per month, or 1 per cent, for two months or 60 days. ORAL EXERCISES. 1. The interest for 60 days 2. " " " 60 " 3. " " " 60 " 4. " " " 60 " 5. " " " 60 " 6. " " " 60 " 7. " " " 60 " 8. u " " 60 " nt (\C/ , cLL \J yfl , u 6^ " 6&/ " 6 C / " / " 6^ " *; on $20 =? $50 =? " $100 =? $75 =? " $125 =? " $175 =? " $200 =? " $316.50=? 9. " " " 60 <( " ^9^ " $715 =? 10. " " " 60 " " 6^! " $50.50=? 11. " " " 60 " " *%> " $215.15=? 12 u u " 60 " " 6% ' " $1000 =? *In New York interest is usually reckoned for the full year. 144 NELSON'S COMMON-SCHOOL ARITHMETIC. 13. The interest for 60 days, at 6%, on $976.14=? 14. " 60 " 6%, $715.15=? 15. " " 60 " " 6%, " $5000 =? 16. " " " 60 " " 6%, " $5 ==? 17. " " " 60 " " 6%, " $23.13=? 18. " " 60 " " 6%, $67.15=? 19. On $7.50=? 28. On $269.14=? 20. " $8,75=? 29. " $198.97=? 21. " $118.67=? 30. $267.18=? 22. $368.56=? 31. $1365.50=? 23. " $210.33=? 32. c< $316.18=? 24. " $67.67=? 33. " $215.16=? 25. $39.37=? 34. " $716.16=? 26. " $21.37=? 35. " $317.60=? 27. " $116.16=? 36. " $167.37=? 157. Having the interest for 60 days, the interest for any shorter time may be found by ALIQUOTS OF 60. 30 days=l 12 days= i 5 day 3=^ 2 days=^ 20 ' = 10 = J 4 =^ 1 day =^ 15 =1 6 = T V 3 = 5 V When the number is not an aliquot of 60, For 7 take 6 and 1 For 29 take 1 off 30 " 8 " 6 " 2 35 " 30 and 5 " 14 12 <c 2 " 38 " 30, 6 and 2 " 19 15 " 4 43 30, 12 1 " 26 " 20 " 6 " 45 15 oif 60 " 27 " 15 " 12 85 add 20 and 5^ *The Teacher should continue these exercises till his scholars are familiar with all the aliquots, and the method of resolving other . numbers into aliquots. SIMPLE INTEREST. 145 1. Fin.d the interest cm $500, for 30 days, at 6% per annum. 3Q=)5.QQ=lHterestfor 60 days. 2M=Interest for 30 days. 2 to 8. Find the interest on $200 for 12 days, 15 days, 20 days, 10 days, 6 days, 3 days, 1 day, at 6% per annum. AHS. $2.23.3. 9 to 13. Find the interest on $160 for 1, 2, 3, 4 and 5 days, at 6% per annum. Am. $0.39.9+. 14 to 18. Find the interest on $240 for 6, 10, 12, 20 and 30 days, at 6% per annum. Ans. $3.12. 19 to 22. Find the interest on $1000 for 1, 10, 12 and 6 days, at 6% per annum. Ans. $4.83.3+. 23. Find the interest on $675, for 27 days, at 6% per annum. $6.75 =Iutcrest for 60 days. 15=1= 1.687 12=4- 1?5_ 3.037 or $3.04. Principal. Time. Principal. Time. 24. $250 for 20 ds=? 30. $650 for 35 ds=? 25. $567 for 14 ds=? 31. $980 for 80 ds=? 26. $968 for 25 ds=? 32. $216 for 93 ds^? 27. $846 for 33 ds=? 33. $800 for 67 ds=? 28. $610 for 18 ds=? 34. $915 for 44 ds=? 29. $918 for 27 ds==? 35. $1200 for 93 ds ? Answers: $16.803+ and $54.448+. 158. Merchants or bankers seldom reckon interest on cents. When under 50 they are rejected; otherwise, a dollar is added to the dollars. It should also be observed that business men express their results in dollars and cents, to which usage the learner ought to conform. For practice or review, the 13 146 NELSON'S COMMON-SCHOOL ARITHMETIC. 51. teacher may require the exact answers instead of those given. 40. $1799.14 for 93 ds=? $3$7.66 for 67 ds=? $199.44 for 41 ds=^? $450.22 for 29 ds^? Total, $35.75. 48. $1997.00 for 13 ds=? 49. $7.88 for 54 dsr^? 50. $17.97 for 35 ds=? $10.00 for 120 ds^? Total, $4.71. 58. $1999.20 for 23 ds=? 59. $361.74 for 18 ds=? 60. $78.93 for 23 ds=? 61. $1467.20 for 34 ds=? 62. $7100.18 for 77 ds=? 63. $29.00 for 99 ds=? Total, $108.96. 159. To compute 6% interest for any number of months. When used in drawing notes or drafts, the month is al- ways calendar; but when computing interest, 30 days is considered a month; hence, the note on page 142 would draw interest for one day more than the three months, exclusive of the days of grace, there being 94 days be- tween the date and the maturity.* This is a very important distinction, as will be seen by reference to Bank Discount. 160. Since there are half as many 60 days as months, we multiply the interest for 60 days by half the number of months. 36. $1000.00 for 27 ds=? 37. $71.97 for 47 ds^=? 38. $61.80 for 45 ds^? 39. $190.27 for 16 ds=? Total, $6.04. 44. $719.99 for 11 ds=? 45. $55.18 for 9 ds=? 46. $88.17 for 69 ds^? 47. $466.00 for 78 ds^? Total, $8.47. 52. $1000.00 for 97 ds=? 53. $650.00 for 67 ds=? $10.70 for 13 ds=? $127.57 for 51 ds=? $368.17 for 118 ds^? $718.57 for 125 ds=? -Total. $46.76. 54. 55. 56. 57. *ln practice, when the note remains unpaid till maturity, interest would be charged for only the three mouths, as if it were 90 days. SIMPLE INTEREST. 147 64. Find the interest on $620, for 4 months, at 6% per annum. Q.^= Interest for 60 days. 2 Number of 60 days in 4 months. \2AQ=Interest for 4 months. Find the interest, at 6% per annum, on the following: 65. $750.25 for 6 mos=:? 69. 910.70 for 5 mos=? 66. $218.87^ for 8 inos=? 70. $876.33^ for 4 inos=? 67. $1000.00^ for 7 mos=? 71. $937.79 for 3 mos=? 68. $560.374 for 9 mos=? 72. $168.00 for 1 ino ==? Answers: $91.46 and $55.21. 73. Required the interest on $350.25 for 7 mos. 15 ds* $3.50 =Interest for 60 days. &k=litumber of 60 days in 7 months. 10.50 1.75 875=Interest for 15 days or J of 60. $13.125" or $13.12 2 . Another way: $3.50 ^Interest for 60 days. 14.00 = Interest for 8 months. 87 '5= Interest for 15 days off. 13.125 or $13.12*. Compute the interest at 6% pr. an. on the following: 74. * $36.57 for 3 mos 20 ds. 78. $1673 for 8 mos 8 ds. 75. $2977 " 6 " 16 " 79. $936 " 4 " 19 " 76. $9856 " 4 " 15 " 80. $281 " 3 " 27 " 77. $2836 " 9 " 27 " 81. $106.27 " 8 " 16 lt Answers: $460.06 and $103.40. *When the principal is small and the time long, interest may be computed on cents. 148 NELSON'S COMMON-SCHOOL ARITHMETIC. Find the amounts of the following, and be particular to add the cents of the principal : Mos. Days. Mos. Days. 82. $250.15 for 6 2 86. $501.19 for 6 6 83. $380.67 " 10 10 87. $219.12^ " 4 27 84. $900.90 " 19 3 88. $369. 16f 8 12 85. $216.67 " 8 20 89. $1220.00 10 10 Answers: $1871.11 and $2408.91. 161. To find the interest for years at 6% per annum. Business usage allows only 360 days to the year, which is 6 sixty days; hence, the interest for 60 days, multiplied by 6 times the number of years, gives the result. 90. Find the interest on $120 for 1 year, 4 months and 20 days, at 6% per annum. 1.20 EXPLANATION. The interest for 60 days is 120 cents; 8 for 1 year and 4 months it is 8 times 120 or 960 cents; q P~ and for 20 days it is J of 120, or 40 cents, making the ' sum $10.00 the interest required. $10.00 Ans. 91. Find the interest of $240 for 3 years, 4 months and 10 days. Ans. $48.40 92. What is the interest of $1467.45 for 2 years, 6 months and 17 days? Ans. $224.21. Find the interest of the following: 93. $321.00 for 2 years 3 months 15 days.* 94. $1767.00 for 7 years 4 mouths 21 days. 95. $897.25 for 3 years 6 months 27 days. 96. $898.57 for 2 years 7 months 25 days.f 97. $716.27 for 2 years 1 month 9 days. Answers: $90.57, $44.14, $783.66, $192.41, $143.09. *Find the interest for 2 years 4 months, and deduct the interest for 15 days. tCall this 2 years 8 months, and deduct the interest for 5 days. \ SIMPLE INTEREST. 149 Find the interest on the following : 98. $810.98 for 1 year 6 months 7 days. 99. $50.00 for 9 years 7 months 18 days. 100. 8.00 for 9 years 3 months 27 days. Answers: $90.58, $73.94, $10.90, $4.48. 101. $3140.79 for 1 year 7 months 7 days=? 102. $795.17 for 2 years 1 month 1 day =? 103. $3.90 for 3 years 5 months 15 days=? 104. $1057.57 for 1 year 11 months 11 days^? Total, $526.01. 105. $2674.57 for 1 year 8 months 21 days=? 106. $7143.45 for 2 years 1 month 18 days=? 107. $1742.67 for 1 year 9 months 13 days=? 108. $2100.00 for 2 years 1 month 1 day ==? 109. $4109.85 for 1 year 6 months 17 days=? Total, $2022.35. 110. $7856.00 for 1 year 1 month 29 days? 111. $677.19 for 3 years 3 months 3 days=? 112. $287.17 for 1 year 7 months 16 days=? 113. $97.19 for 5 years 10 months 14 days=? 114. $10.10 for 1 year 3 months 19 dys=? Total, $743.95. 115. $57.87 for 2 years 6 months 14 days=? 116. $120.14 for 7 years 7 months 7 days=? 117. $340.00 for 9 years 1 month 24 days=? 118. $1657.00 for 1 year 3 months 24 days=? 119. $769.75 for 2 years 3 months 18 days=? Total, $487.40. . : . The Teacher, when reviewing these exercises, may require his class to compute interest on cents, as is usually done in courts of justice. 150 NELSON'S COMMON-SCHOOL ARITHMETIC. 162. Having the interest at 6% per annum to find it at any other rate, all-quo ts of 6 may be used. At 1% it will be of that at 6%. At 5% it will be -J- less than at 6%. At 7% it will be -|- more than at 6%. At 2% it will be of that at 6%. At 4% it will be 1 less than that at 6%. At 8% it will be J more than that at 6%. At 9% it will be more than that at 6%. At 10% it will be ^ of that at 6%. 120. Find the interest on $250 for 1 year, 3 months and 20 days at 7% pr. annum. $2.50 Intercut for 00 days at 6%. ? I, = Number of GO days. 125 " 1750 20 <Zoya. a 6%, 3.263=/ttteresl at 1%. $22.8S=Interest at 7%. or $22.85. 121. 708.18 for 6 yrs 1 mo 6 ds 9% pr. an.=? 122. $1000.00 for 4 yrs 2 mos 4 ds @ 7^ pr. au.r=? Answers: $438.10, $292.44. 123. $340 for 2 yrs 2 mos 20 ds @ 2J% pr. an.=? 124. $000 for 3 yrs 4 mos 15 ds @ 6%% pr. an.= ? 125. $850 for 1 yr 2 mos 12 ds at \% pr. an.=? Total, $237.22. Find the interest of 126. $617.18 for 3 mos 18 ds @ 15% pr. an. 127. $460.74 for 2 mos 5 ds @ 18% pr. an. 128. $765.12 for 8 mos 16 ds @ 20% pr. an. Total, $151.55. SIMPLE INTEREST. 151 Find the interest on the following at 10% per annum: 129. $710 for 92 days, 133. $496 for 91 days. 130. $19^8 for 27 days. 134. $671 for 80 days. 131. $8889 for 128 days. 135. $100 for 104 days. 132. $75 for 117 days. 136. $269 for 73 days. Total, $351.47. Total, $36.91. 163. It is customary for bankers to lend money, and discount by the month instead of the year. This percent- age is easily converted into 6% interest, and the work performed with as much ease as before. 1 % per month is 12% per year, or 2 times 6%. l-J-% per month is 18% per year, or SHimes 6%. 2 % per month is 24% per year, or 4 times 6%. Find the interest on the following: 137. $65 for 80 days @ 2 % per month. 138. 840 for 33 days @ \\% per month. 139. $190 for 63 days @ 2 % per month. 140. $700 for 93 days @ 3 % per month. Total, $77.20. Find the amount of the following 141. $710 for 36 days @ l-J-% per month. 142. $216 for 45 days @ 2 % per month. 143. $1800 for 57 days @ lJ-% per month. 144. $560 for 14 days @ 6J% per month. Total, $3367.89. When computing interest, the ingenious student will contrive many ways for abbreviating his work. Some- times he will take advantage of the aliquots of 100; at other times he will transpose the terms, and consider the days as dollars and the dollars as days, or he will reduce the rate mentally to 6%, if.it is some other rate, and thus simplify as well as abbreviate. For instance, in the ques- 152 NELSON'S COMMON-SCHOOL ARITHMETIC. tion 138, lie might consider the $40, 120, and transposing the terms, which could be done instantly, he would simply have to multiply the 33 by 2, making 66c, the answer. It will materially abridge the operation and expedite the labor, if the learner will observe to avoid the use of all lines, figures or marks that arc not absolutely necessary. As, for instance, when using aliquot parts, to write only the results of division, as shown in the following example: 145. Interest of $321 for 2 years, 1 month 3.21 and 22 days at 10% per annum. 40 125 EXPLANATION. Mentally it is found that there are 12 J 1.07 60 days in 2 years and 1 month, to multiply by which we 107 divide by 8. The division by 6 and the multiplication ji 0^9 by 10 were performed simultaneously, giving $08.836 or $08.84 as the answer. 68.836 In Bank Discount, the learner will find numerous ques- tions upon which to exercise his ingenuity, arid the judi- cious teacher will encourage him. The method of finding True Discount, and other more dif- ficult calculations, will be found at the end of Banking. PARTIAL PAYMENTS, OR PAYMENTS BY INSTALLMENTS. 164. Notes, bonds, etc., drawing interest, are sometimes paid by installments, and the amounts thus paid indorsed on them. The legal rule for computing interest on install- ments may be expressed thus: Apply the payment to the discharge of the interest, and if there is a remainder, subtract it from the debt. When the payment is less than the interest due, it is not applied to the discharge of the interest or debt, but is indorsed on the note until the installments exceed the interest. The sum of the installments is then taken from the amount due, and interest computed on the remainder as before. SIMPLE INTEREST. 153 1. $576. CINCINNATI, Oct. 9, 1857. On demand, I promise to pay Robert Ingles, or order, Five hundred and seventy-six dollars, with interest. Value received. SAMUEL DUNNING. On the note are the following indorsements : Rec:d, Dec. 16, 1857, $100. " Pel. 28, 1858, 3. " July 27, 1858, 150. Required the amount due September 3, 1858. Yrs. Mos. J)s. From 1857 12 16 Take 1857 10 9 Difference, 2 7 or 67 days. $576.00 ^4 wowntf of note. 6A3=Intercst on $576 for 67 days. $582.43= Total amount due. to be subtracted. 2.43= Balance due. The second payment is less than the interest due and no calculation is required. From December 16, 1857, to July 27, 1858, is 7 months 11 days. $482A3=:Balance due. 17 .! Interest for 7 mos. 11 days. $500.18 .^moimtf due. 153.QQ= Amount of payments, $347 '.18= Balance due. From July 27 to September 3, is 38 days. 2.W=Literest for 38 days. due September 3, 1858. 154 NELSON'S COMMON-SCHOOL ARITHMETIC. 2. $650. BOSTON, June 3, 1848." For value received, I promise to pay, on demand, to H. Crooks, or order, Six hundred and fifty dollars, with interest at 6% per annum. J. F. DAVIS. Indorsements. Jan. 6, 1850, $95. Oct. 13, 1850, 350. June 3, 1855, 12. Sept. 7, 1857, paid the balance; how much was it? Aris. $405.92. 3. On a note drawn September 23, 1857, for $650, with legal interest, there are the following indorsements : Oct. 4, $100. Nov. 3, 2. Dec. 11), 210. April 3, 1858, the balance; how much was it? Am. $354,32. 4. On a. note drawn October 3, 1856, for $1000, with interest, are the following indorsements: Nov. 30, $50. Jan. 6, 1857, 100. Feb. 9, " 5. June 6, " 190. Feb. 3, 1859, the balance; what was the amount? XVIII. BANKING. 164. Banking is the business of dealing in money. Banking houses borrow and lend money, receive money for safe keeping, and exchange the money of one country for that of another. There are National Banks, Public and Private Banks, Banks of Deposit, Banks of Issue or Circulation and Banks of Discount. BANKING. 155 A National Bank is one which issues notes secured by Donds of the United States, deposited with the United States Treasurer, and is doing business under the authority of the General Government. A Public Bank is one that is owned by a joint stock company, who commit its management to some of their number chosen for that purpose. These persons are called President and Directors. A Private Bank is one which is owned by one or more individuals, who attend to its business personally. Banks of Deposit receive the ready money of merchants and others for safe keeping. Banks of deposit also loan money on interest and some pay interest on deposits. Banks of Issue or Circulation manufacture and issue paper money, called bank-notes or bills. Many of these batiks also receive money on deposit and do a discount business. Banks of Discount lend money on interest, when suita- ble security is given. The security required by banking institutions when loaning money, is a note or notes from the borrower, with the names of one or two responsible persons written on the back, in such a manner as to bind them for payment, should the drawer of the note fail to pay it at the proper time. The persons who sign their names thus are called indorsers, the writing the indorsement, and the person who writes the note and signs his name, the maker or drawer. The notes given by borrowers may be their own* or * Notes are always known by the names of the makers. We Bpeak of A B 1 s note, though it is in our possession. To hold an indorser, he must be notified of the maturity of the note at furthest on the next business day, if he resides in the same town or city ; otherwise, a notice should be mailed to him within the same time. 156 NELSON'S COMMON-SCHOOL ARITHMETIC. those of other persons. If those of others, the borrower's indorsement alone is all that is usually asked. The person to whom the note is made payable is called the payee, and the person possessing it, the holder. By the face of the note is meant the side containing the prom- ise; also the amount for which it is drawn. A negotiable note is one that can be transferred. To be negotiable, it should be drawn "to the order' 1 of the payee, or to him " or order" or "to bearer." In the first two cases, to make it negotiable, the payee would have to write his name on the back. To the third class belong bank-notes, whic^ may be passed by any holder without indorsement. The phrase "value received" is not considered essen- tial to a note, though it is well to insert it. Days of Grace are three days allowed to the maker of a note beyond the stated time of payment. A note that is drawn payable 90 days after date, is not legally due until the 93d day. 165. MATURITY OF NOTES. Notes falling due on Sunday or a national holiday are payable the day before. A note dated the 28th, 29th, 30th or 31st of Jan- uary, at one month, falls due on the last day of February without, or on the 3d day of March with grace ; and a note dated the last day of February, at one month, falls due on the 28th of March, if not a leap-year ; otherwise, on the 29th of March, plus the days of grace. Hence, a note drawn by the month, and dated on the last day of the month, falls due on the same day of the month, if the latter month have a corresponding day; otherwise, it falls due on the last day of the month. Thus, a note dated BANKING. 157 November 30th, at two months, without grace, would fall due on the 30th of January ; at three months, oil the 28th or last day of February. 166. REMARKS ON NOTES. 1. A note need not be dated at the place where it is drawn, and can be made payable at any particular place the parties may agree upon; but to hold an indorser, de- mand must be made at the particular place specified. A note is also good if dated on Sunday. 2. When giving a note, the maker ought to fix the place Df payment say his bank or place of deposit if the payee be agreeable. 3. In New York and other States, notes draw seven per cent, interest. Notes made payable in those States, accord- ingly, draw that rate of interest after maturity. 4. If made by more than one person, it is called a joint note, or joint and several. It is a joint note only, unless words are used to indicate individual responsibility, as "We jointly and severally," etc. 5. It is customary for merchants to deposit their notes in bank for collection, the bank, by this means, undertak- ing to see them paid, or using means to hold the indorsers. 6. A note drawn payable to the maker, and by him in- dorsed in blank, can be transferred negotiated without liability to the holder. 7. When taking up a note paying it the drawer ought to require the indorsement of the holder, deface his own signature and file the note away, as it may subserve the purpose of a receipt at some future time. 8. A note drawn under the seal of the maker is called a bond. Indorsements are of various kinds, and, like the note itself, they require no stated form of words. 158 NELSON'S COMMON-SCHOOL ARITHMETIC. There are Blank, Full, Special and Restrictive indorse- ments. A Blank Indorsement is made by the payee or holder writing his name on the back. A Full Indorsement, or, as it is called, an indorsement in full, is made by writing such a transfer as "Pay to the order of A," or words to that effect, and signing the name. A Special Indorsement is one made to suit a particular case, as when the indorser wishes to free himself of re- sponsibility should the maker fail to pay the note. "Pay the the contents to B, or order, without recourse on me." "Payee." A Restrictive Indorsement restrains the pa} r ment of the note to the party to whom it is indorsed, as "Pay the con- tents to C only." This kind of indorsement does not limit the payment when made by another person than the payee of the note. The payee is called the first indorser, if he transfers the note, and the party to whom he transfers it, the indorsee. Should the latter again transfer it, he would be called the second indorser and the party to whom it was transferred, the second indorsee. Indorsements ought to be written across the back, with the left end up, as shown on page 160. 167. PtEMARKS ON INDORSING. 1. A holder or indorsee can write over a blank signature a full or restrictive in- dorsement. 2. A holder may cancel all indorsements (signatures) but the first, and may cancel a full indorsement, except the signature of the first. 3. The object of a full indorsement is to prevent its transfer without the signature of the holder. 4. A note may be transferred after it is due, but the indorser is not then liable. BANKING. 159 5. The payee or his agent must make the first transfer. 6. Any partner of a firm may indorse for all. After maturity, it ought to be done severally. 7. Agents may indorse for principals thus: A B, Principal, By C D, Agent. 8. An indorsement cancelled by mistake does not dis- charge the indorscr. FORMS OF PROMISSORY NOTES. The following are given as examples for the learner to to copy. The printed blanks differ from these, and are easily filled. A NEGOTIABLE NOTE. CLEVELAND, Sept. 30, 1866. Thirty days after date, I promise to pay to the order of J. C. Hutsiupiller Five hundred sixty T 3 ^j- dollars, at the First National Bank, this city, value received. $560^. E. R. FELTON. A NON-NEGOTIABLE NOTE. MILWAUKEE, Apr. 4, 1867. Ninety days after date, I promise to pay to Gr. W. Nel- son One thousand dollars, value received. $1000. R. C. SPENCER. A JOINT AND SEVERAL NOTE WITH INTEREST. CINCINNATI, Aug. 9, 1867. On demand, six months after date, we severally and jointly promise to pay to the order of William Otte Six thousand dollars, value received, with interest at six per cent, per annum. GEORGE F. SANDS, W. J. BREED. 160 NELSON'S COMMON-SCHOOL ARITHMETIC. FORM OF INDORSEMENT. When writing the following notes, the learner ought to use cap paper, so that the notes be long enough for filing, and leave a margin above and below of not less than one and a half lines. HOME EXERCISES IN DRAWING NOTES. Draw notes from the following data: 1. Date, Aug. 18, 1868. Place, Louisville. Maker, Yourself. Time, On demand. Face, $678.14. Payee, S. E. Peyton. 3. Date, April 9, 1870. Place, Pittsburgh. . Drawer, S. S. Sargent. Payee, Yourself. Face, $168.75. Time, 3 months. 2. Date, Nov. 6, 1869. Place, Cincinnati. Drawer, Yourself. Payee, James Moore. Fnce, $375.50. Time, 90 days. 4. Date, Dec. 3, 1869. Place, Louisville. Time, On demand. Drawer, E. P. Bowers, Payee, Indorser, R. Johnson. Face, $6745.15. BANKING. 161 5. Date, Oct. 19, 1860. 6. Date, Place, Wheeling, Va. Drawer, Archibald Warren. Payee, Yourself. Indorser, Face, $1364. Time, 6 months. Indorsee, Saml. Adams. Nov. 29, 1867. Date, Place, Time, Face, Drawer, Payee, 1st Indorser, , Indorsee, Yourself. April 3, 1869. New York. 4 mouths. $1408.75. James Todd. Robt. Emmet. 9. Place, Date, Time, Maker, Payee, 1st Indorser, James Moore. 2d Indorser, Yourself. Face, Mobile. 10. April 3, 1867. One day afterdate. S. 0. Miner. 11. Place, Boston, Mass. Date, July 10, 1870. Time, 30 days. 1st Indorser, Robt. Penn, 1st Indorsee, James Moon. Place, New Orleans. Drawer, E. M. Small. Payee, H, C. Piner. 1st Indorser, 1st Indorsee, Edward Epply. 2d Indorser, Face, $138. Date, Jan. 1, 1867. Place, Chicago. Time, 60 days. Drawer, R. E. Baird. Payee, J. E. Piner. Indorser, Indorsee, J. J. Wood. Face, $1637.00. Place, San Francisco. Date, Time, 4 months. Maker, Henry Adams. Indorsee, Thomas Orr. Payee, Henry Adams. Indorser, Face, $450. Maker, S. Poor. Payee, 2d Indorser, ..* 2d Indorsee, Wm. A. Miller. Face, $396.57. Draw the notes necessary for the following transac- tions : 12. Take H. P. Spike's obligation, at 1 day after date, for price of a house worth $1695.* *To make this a bond, insert the word "seal" after the name 14 162 NELSON'S COMMON-SCHOOL ARITHMETIC. 13. Pay S. 0. Mooney 167.87 with your note at 30 days, bearing interest. 14. Lend Henry Shotwell 1687, on his note, in such a way as to enable you to collect the money at any time. 15. Jno. Emerson's note, our favor, indorsed to G. Young, for 675 at 90 days, was protested for non-pay- ment on January 3, 1866. What was its form? 16. Renew your note of $190, and take 3 months' time, allowing interest for the time the new note has to run. 17. Pay Juo. Sallust 367, on acct., with a note at 90 days. 18. Indorse C D's note of Sept, 3, I860, for 6600 at 90 days (your favor), in such a way that you will not be held responsible for payment. 19. Give W. A. Moore your note at 3 months, and draw it in such a way as will prevent him from negotiat- ing it. 20. Exchange notes with your neighbor, each drawing at 90 days, and indorse. 21. Draw a copy of your note for 675, favor of H. Corncob, at 3 months, upon which you had to pay $2.47 interest on Sept 19, at 6%. 22. Pay Saml. Saul Pickwick the balance of S678.50, you owe him, by giving note in favor of Sam. Weller at 60 ds. 23. Take the note of J. Morgan Henry, Madison, Ind., for $360, at 3 months, and secure yourself against the statute which allows " relief," etc. (After the words "value received," insert "without any relief whatever from valuation or appraisement laws." For form of note, see page 202.) 24. Indorse the receipt of 167.87, amount paid you on E. C. Johnston's note, dated Aug. 9, 1866, and drawn for 974.35 at 6 months, your favor. BANKING, 163 CHECKS DRAFTS BILLS OF EXCHANGE. A Check is a written order on a bank or its cashier for the payment of money in its possession belonging to the party making it. A Draft* is similar to a check, but is written more formally and may be drawn upon any other than a banker; and a Bill of Exchange is a check or draft used for transmitting money to distant places. A check, like a promissory note, may be drawn to bearer, to order, or be made payable to a particular per- son. When given to strangers, or for large sums of money, they should be drawn to order, so that those receiving them would have to indorse them before payment. In this way, they may also be used to subserve the purposes of receipts. Checks may be antedated or post-dated. In the former case, they are payable on presentation, if there are funds of the drawer to meet them; in the latter, they will not be payable till the date arrives. Should a bank refuse to pay a check, the party to be sued is the maker, or trans- ferrer, if it has been transferred. If a bank pay a forged check, the bank, and not the person whose signature is forged, has to sustain the loss, though the forgery may be so well executed that it caa not be detected by ordinary inspection. When a check is so carelessly drawn that an alteration may be easily made, the loss arising must be borne by the drawer. For instance, the amount to be paid should commence at the extreme left of the line, and the part left unfilled be written with a curved line or other mark to prevent additions. *For forms of draft, see Exchange. 164 NELSON'S COMMON-SCHOOL ARITHMETIC. FORM OF CHECK. OUDERS DUE-BILLS. A mercantile Order is a request for the delivery of goods or money. It differs from a draft in being more simple in its form, and its being usually drawn for the payment of goods instead of money. The following is the common form: CINCINNATI, Jarfy 1, 1867. MESSRS. JOHN SHILLITO & Co. Will please let Samuel Steele, or bearer, have goods to the amount of One hundred dollars, and charge to my ac- count. $100^. GEORGE KIDD. To be more specific, the kind of goods and the price, wholesale or retail, might be inserted. A Due-bill is simply an acknowledgement of a debt. It is usually drawn for a small sum and to settle an ac- count. Due-bills are considered to draw interest from date, though it is seldom exacted. BANKING. 165 FORM OF A DUE-BILL. INDIANAPOLIS, Aug. 3, 1868. Due Henry Ellsworth, or order, Fifteen dollars. Value received. $15_QLQL > HENRY SPENCER. RECEIPTS. It may not be considered improper here to introduce a few forms of receipts, as they are essentially connected with the preceding subjects. A Receipt should specify what it was given for, whether money, goods, note, etc., the amount for which it was given and the date, and the amount should be in writing. Receipts for sums over twenty dollars should have a stamp. Receipts may be made on bills, notes, etc., or given separately, and need not be of any particular form. RECEIPT ON ACCOUNT. CINCINNATI, Jan. 1, 1868. Received of Mr. John Cummins, One hundred and twenty-five dollars and 23 cents on account. -. JAMES MORGAN. RECEIPT FOR MONEY ON NOTE. PITTSBURG, Apr. 3, 1867. Received of Alex. Cowley, One thousand dollars, to be credited on his note, my favor, dated Jan. 3, 1867, for six thousand dollars. $1000. W. ALLAN MILLER.* *A payment on a note should be receipted (indorsed) on the back of the same, and a statement made that a receipt was given. 166 NELSON'S COMMON-SCHOOL ARITHMETIC. RECEIPT IN FULL. Received, Cincinnati, June 3, 1SG8, of Timothy Hay, Thirty -five -,YV dollars, in full to date. $35.25. THOMAS E. YOUNGMAX. RECEIPT FOR RENT ON ACCOUNT. BOSTON, Feb. 27, 1867. Received of Mr. Henry G. Judkins Sixty-eight dollars, on account of rent for No. 6 Long St. $68.00. EDW. FABER. RECEIPT FOR A NOTE. PHILADELPHIA, Jariy 30, 1869. Received of Mr. James Thompson, his note, my favor, at ninety days, for Five hundred dollars to balance acct. $500. JNO. F. GREEN, JR. RECEIPT FOR MONEY IN ADVANCE. COLUMBUS, Apr. 3, 1869. Received of J. Q. A. Miller, Forty dollars, in advance, for live pork, to he delivered to him on or before October 1 1869, at 7 cents a pound. $40. JAS. HOLLANDER, SEN. RECEIPT FOR MONEY RECEIVED ON ACCOUNT OF ANOTHER. CINCINNATI, Oct. 10, 1866. Received from H. D. Brown, on the account II. W. Sou they, Sixty-nine y 8 ^ dollars, in full of acct to 1st inst. $69^. Y. W. COOK, JR. RECEIPT FOR RENT IN FULL. NEW YORK, Sept. 12, 1866. Received from R. Quinn, Twenty-five dollars, in full for rent of house, No. 96 Chestnut St., to 9th iust. $25. JAS. M. HALL. BANKING. 167 RECEIPT FOR MONEY RECEIVED BY A CLERK. NEW YORK, Aug. 3, 1870. Received of II. Simon One hundred dollars on acct. . JAMES MOORE. Per JNO. WOOD, Clk. HOME EXERCISES IN DRAWING RECEIPTS. 1. Receipt to John Roberts for $100 on account. 2. Give H. C. Parker a receipt for $50 in full of ac- count. 3. Draw a receipt for $250 T 5 ,?g- in favor of C. C. Martin, for his note of this date in settlement of account, and draw a copy of the note, 4. Receipt to Mrs. T. H. Henshaw for $365, in part for rent of house, 1968 Vine Street. 5. Give your teacher a receipted bill for 257 barrels of flour at $12.67 per barrel; drayage, $8.50. 6. Let William Eduiuridson have your receipt for $67, on his note, of the 3d of last month, at 6 months, your favor; and draw a copy of the note, showing a receipt to be made on it. Face of note, $936. 7. M*ke out to your teacher, this date, the bill on page 111, and take a note in settlement. 8. Receipt for $150 cash on the first bill, page 112, making it to William Nelson. 9. As clerk of II. J. Estcourt, make out the first bill on page 113 to your teacher, allowing him a discount of 5% for cash. 10. Give your due-bill and an order on your grocer, to teacher, for tuition one session of 5 months, amounting to $85. Make the order for a balance of $15.75 in grocer's hands, and draw all the papers. 168 NELSON'S COMMON-SCHOOL ARITHMETIC. TIME TABLE FOR COMPUTING INTEREST AND AVERAGE. Number of days from Is/ of Jan-nary to any other day of the year. In leap-years, add I to the days after 26th of February. a o i j.Tiinmry .... I jFobrujir; *i * T3 g S X -l S3 3 0> ,Day of Mo. ^ e^ 5T I August cc 4 3 1 I n 1 1 November i December. I! Day of Mo. i : 1 : 1 :51 59 90:120 151 1 181 212j243|273 304 334 1 2 1 32 60 91 121 152 2 182 213 244i274 305 335 2 3 2 33 61 92 122 153 3 183 214 245 275 306 336 3 4 3 34 62 93 123 154 4 184 215 246 276 307 337 4 5 4 35 63 94 124 155 5 185 216 247 277 308 338 5 6 5 36 64 95 125 156 6 186 217 248 278 309 339 6 7 C 37 65 96 126 157 7 187 218 249 279 310 340 7 8 7 38 86 97 127 158 8 188 219 250 280 311 341 8 9 8 3! 67 98 128 159 9 189 220 251 281 312 342 9 10 9 40 68 99 129 160 10 190 221 252 282 313 343 10 11 10 41 69 100 130 161 11 191 222 253 283 314 344 11 12 11 42 70 101 131 162 12 192 223 254 284 315 345 12 13 14 12 IB 43 44 71 102132 721031133 163 164 13 14 193 194 224 225 255 256 285 286 316 317 341) 347 13 14 15 14 15 73 104 134 165 15 195 226 257 287 318 348 15 16 15 46 74 105 135 166 16 196 227 258 288 319 349 16 17 1(5 47 75 106 136 167 17 197 228 259 289 320 350 17 18 17 48 76 107 137 168 18 198 229 260 290 321 351 18 19 18 49 77 108 138 169 19 199 230 261 291 322 352 19 20 19 50 78 109 139 170 20 200 231 262 292 323 553 20 21 20 51 79 110 140 171 21 201 232 263 293 324 354 21 22 21 52 80 111 141 172 22 202 233 264 294 325 355 22 23 22 53 81 112 1421173 23 203 234 265 295 326 356 23 24 23 54 82 113|143 174 24 204 235 266 296 327 557 24 25 24 55 83 114 144 175 25 205 236 267 297 328 -558 25 26 25 56 84 115 145 176 26 206 237 2681298 329 359 26 27 26 f)7 85 116 146 177 27 207 238 269 299 330 360 27 28 27 58 86 117 147 178 28 208 2391270 3001331 561 28 29 28 87 118 148 179 29 2091240 271 301 332 362 29 30 29 88 119 149 180 30 210 241 272 302 333 363 30 31 30 89 150 31 211 242 303 364 31 BANKING. 169 TIME TABLE FOR COMPUTING INTEREST AND AVERAGE. Number of days from 1st of July to any other day of the year. In leap-years, add 1 to the days after 28//t of February. b p s. o fT August , ^September .. I ^ i g. o ^ 1 November ., ! December... Day of Mo... [January I i February.... |March > V " | <*$ -| Bl 'Day of Mo... i 1 31 62 92 123 153 1 1841215 243 274 304 335 1 2 1 32 63 93 124 154 2 185 216 244 275 305 336 2 3 2 33 64 94 125 155 3 186 217 245 276 306 337 3 4 *j 34 65 95 126 156 4 187 218 246 277 307 338 4 5 4 35 66 96 127 157 5 188 219 247 278 308 339 5 5 36 67 97 128 158 6 189 220 248 279 309 340 6 7 6 37 68 98 129 159 7 190 221 249 280 310 341 7 8 7 38 69 99 130 160 8 191 222 250 281 311 342 8 9 8 39 70 100 131 161 9 192 223 251 282 312 343 9 10 9 40 71 101 132 162 10 193 224 252 283 313 344 10 11 10 41 72 102 133 163 11 194 225 253 284 314 345 11 12 11 42 73 103 134 164 12 195 226 254 285 315 346 12 13 12 43 74 104 135 165 13 196 227 255 286 316 347 13 14 13 44 75 105 136 166 14 1971228 256 287 317 348 14 15 14 45 76 106 137 167 15 198 229 257 288 318 349 15 16 15 46 77 107 138 168 16 199 230 258 289 319 J50 16 17 16 fc7 78 108 139 169 17 200 231 259 290 320 351 17 18 17 48 79 109 140 170 18 201 232 260 291 321 352 18 19 18 49 80 110 141 171 19 202 233 261 292 322 353 19 20 19 50 81 111 142 172 20 203 234 262 293 323 i&4 20 21 20 5182J112 143 173 21 204 235 263 294 324 355 21 22 21 52 831113 144 174 22 205 236 264 295 325 356 22 23 22 53 84lll4 145 175 23 206 237 265 296 326 }57 23 24 23 54 85 115 146 176 24 207 238 266 297 327 358 24 25 24 55 86116 147 177 25 208 239 267 298 328 359 25 26 25 5(5 871117 148 178 26 209 240 268 299 *29 J60j 26 27 26 57 88 118 149 179 27 210 241 269 300 330 '361 27 28 27 58 89 119 150 180 28 211 242 2701301 331 362 28 29 28 59 90 120 151 181 29 212 271!302 332 '363 29 30 29 60 91 121 152 182 30 213 272 303 333 364 30 31 30 61 122 183 31 214 273 334 j SI 15 170 NELSON'S COMMON-SCHOOL ARITHMETIC USE OF THE PRECEDING TABLES. 1. From ~[st of July to 9//i of June, how many days? Opposite 9, in the last column, is 342, the answer. 2. From 2d of October to 17th November, liow many days? Opposite 2, in the middle column, is 93, in October column, and opposite 17 is 139, in the November column. The difference is 46. These tables are specially adapted for averaging ac- counts, taking 1st of January and 1st of July as dates from which to reckon. DISCOUNTING NOTES. 168. Discounting notes consists in buying them at less than their nominal value, or the amount for which they are drawn. The difference between the nominal value and the price paid is called discount. 169. Bankers prefer lending money on short time, and by the day, instead of by the month. Notes are usually drawn for 30, 60 or 90 days; and interest is always charged on the days of grace. 170. There are two kinds of discount: True Discount, which is the interest paid in advance on the present value of a note, and Bank Discount, which is interest paid in advance on the face of the note. The latter resembles compound interest, as it is interest on both interest and principal.* When a note is discounted in bank, the interest of the note for the time it has to run, and at the banker's rates, is deducted from the sum called for by the note. The bal- *The present worth of a note drawn for $100, payable in a year at 6 per cent., is $94.84, and the interest is $5.56; that is, the prin- cipal and interest together are equal to $100, or the face of the note; BO when a banker discounts from the face of a note, he discounts ou both principal and interest. BANKING. 171 ance is called the proceeds. This species of discount is therefore reckoned in the same way as interest. Bankers reckon interest on every day intervening between the day of discount and that of maturity, including the latter. 171. When a note is drawn by days, subtract the ex- pired term from the number of days for which it is drawn, plus the days of grace ; but when drawn by the month, first find the day of maturity and reckon the whole number of days from the day of discount to that date. 1. Discounted on the day of date, how much discount jhould be deducted from a note of $500 at 90 days?* $5. 00= Interest for GO days. 2.50== " " 30 " .25^ 3 " (grace.) Arts. $7.75 2. $1500. COLUMBUS, Jan. 8, 1859. Sixty days after date, I promise to pay Messrs. M'Ewen and Banfill One thousand five hundred dollars, value received. GEO. K. TENNEY. Required, the discount at 6% per an. Ans. $15.75. 3. $3500. WHEELING, Oct. 3, 1858. Ninety days after date, I promise to pay John M'Culloch, or order, at First National Bank, three thou- sand five hundred dollars, value received. MILO G. DODDS. Required the proceeds at 6% per an. Ans. $3445.75 4. Find the proceeds of a note for $120 at GO days at \% P er month. 5. Required the proceeds of a note dated Jan. 1, 18Q6 J and drawn for $575.75 at 90 days. *Wken the rate is riot named, six per cent, per annum is under- stood. 172 NELSON'S COMMON-SCHOOL ARITHMETIC. 6. What is the bank discount on a note of $450 for 60 days at 2% per month?* 7 to 12. Find the proceeds of the following: On $850 at 30 days at \\% per month; on $1678.25 at 90 days and \\% per mouth; on $670 at 60 days and 2% per month; on $1749.57 at 90 days, 1J% per month; on $688 at 90 days, Z\% per month; on $6784 at 60 days, If % per month. Answers: $118.74, $566.82, $344.22, $18.90, $1668.19, $6534.69, 640.01, $1613.22, $6594.69, $641.86. Find the discount on the following: 13. $1310.00 for 60 days @ 2 % per month. 14. $746.87 for 90 days @ \\% per month. 15. $219.56 for 30 days @ 1 % per month. 16. $1867.25 for 20 days @ 1\% per month. 17. $1367.00 for 15 days @ 3 % per month. Total, $152.57. 18. A note drawn on February 13, 1866, for $900, at 90 days, was discounted on March 23, at 2% per month; how much was paid to the borrower? Am. $867. 19. What proceeds should be paid on a note of $346: at 90 days, drawn on November 3, and discounted on De- cember 7, at \\% per month? Ans. $335.79.' 20. A note for $689, made September 9, payable in 60 days, was discounted on October 5, at 2% per month; what was the discount? Ans. $16.99.i While the cents in the principal are rejected in comput- ing interest and discount, they are always reckoned when finding the amount or proceeds. *Such questions as these may be abbreviated by mentally in- creasing the days or principal in the ratio that the rate is to 6 per cent. In this case, reckon interest on $450X4 or $1800 at G per cent, per annum. BANKING. 173 Required the discount on the following: Face of note. Date. Time. When disc'd. Rate of discount. 21. $167.50 Jan. 3, 1869, 60 ds, Feb. 7, 2 % per mo. 22. $9876.00 Feb. 7, 1869, 90 ds, Mar. 12, 2J% per mo. 23. $789.00 Juu. 18, 1869, 30 ds, July 3, l\% per mo. 24. $1897.00 Feb. 21, 1869, 90 ds, Apr. 1, l|% per mo. Total, $555.24. Find the proceeds of the following : 25. $676.37 Apr. 3, 1869, 90 ds, May 9, 2 % per mo. 26. $679.39 Mar. 9, 1869, 30 ds, Apr. 3, ty% per mo. 27. $7168.00 June 13, 1869, 60 ds, July 9, \\% per mo. 28. $816.37 Aug. 12, 1869, 30 ds, Sep. 6, Z\% per mo. Total, $9172.40. 29. A note for $4378.35, dated February 1, 1867, at 4 months, was discounted on May 16 at 10% per annum; [required the proceeds. SOLUTION. This note falls due June 4. From May 16 to that date is 19 days, giving for discount on $4378, 823.11. Proceeds, $4355.24. 30. Required the proceeds of a note dated September 3, 1867, for $396.87, at 3 mouths, and discounted on Octo- ber 5, at \\% per month. 31. A note dated September 30, at 3 months, and drawn or $1367.56, was discounted on October 1, at 2% per month; what were the proceeds? 32. Required the proceeds of a note dated December 31, 1867, drawn for $5363.75, at 2 months, and discounted February 1, at \\% P er month? 33. A note for $1000, dated Februray 28, 1866, at 2 months, was discounted on March 20, at 1% per month; required the proceeds. Answers: $986.00, $384.56, $1282.75, $5296.70 $529.50, 174 NELSON'S COMMON-SCHOOL ARITHMETIC. Find the proceeds of the following: Fnre of note. Date. Time. When clisc'd. Kate of discount. 34. $2676.00, Jan. 9, 1869, 90 days, Feb. 1, \\% pr mo . 35. 87187.00, Feb. 3, 1869, 60 days, Mar. 13, l%% pr mo. 36. -$768.21, Mar. 6, 1869, 30 days, Apr. 3, 2 % pr mo. 37. $314.00, Apr. 7, 1869, 90 days, May 15, 2 % pr mo. Total discount, $181.95. Ain't of note. Date. Time. When discounted. Rate of disc't. 38. $6785, Dec. 6, 1868, 6 mo. Dec. 29, 2 % pr mo. 39. $3748, Jan. 3, 1868, 5 mo. Feb. 3, 1868, 2% pr mo. 40. $6983, Mar. 9, 1868, 4 mo. June 8, 1868, l\% pr mo. Total proceeds, $16272.70. Amount. "Date. Time. When disc'd. Rate of discount. 41. $3784, May 6, 2 mo. July 3, 2 % per mo. 42. $6987, Jun. 8, 3 mo. Aug. 27, \\% per mo. 43. $7854, July 24, 4 mo. Sept. 17, 1 % per mo. Total proceeds, $18371.58. XIX. TRUE DISCOUNT. 172. TRUE DISCOUNT is the difference between the pres- ent worth of a note and the amount for which it is drawn. The present worth of a note or bill due at a future time without interest, is such a sum as would, if put at inter- est for the same time and rate, amount to the debt; and the difference between this sum and the debt is the discount. Except in courts of justice, this kind of discount is sel- dom used, business men preferring bank discount for its simplicity. DISCOUNT ON INTEREST-BEARING NOTES. Bankers dis- count off the value of the note, including interest, at matur- ity ; while Real Estate Brokers discount off the face of the note, plus the accrued interest, at the time of discount. TRUE DISCOUNT, ETC. 175 1. What is the true discount on a note of $700 for 90 days at 6%? The amount of a dollar for 93 days is $1.0155, by which, if we divide $700, we will find the present worth. OPERATION. $1.0155)700.0000(689.315 60930 NOTE. The interest, on $1 for 90 clays is 90700 0155. The present value of $1.0155, for 93 81240 days is, therefore, $1, and, accordingly, the " qjrn (\ present value of $700 for 93 days is $700 di- 91395 rided by $1.0155 or $089.31, and the discount $700 $689.31, or $10.09. 32050 30465 PROOF. The interest on $689.315, 93 days, 15850 s $10.683, which, if added to the principal, 10155 will give $699.999 or $700. 56950 50775 The pupil can prove his calculations by interest. 2. What is the true discount on a note of $575 for 90 days at 6%? 3. What is the true discount on a note of $137.09 for 90 days at 6%? 4. What will be the proceeds of a note of $1878.67 at 90 days, true discount? 5. A note for $485.44, payable in 30 days after date, is worth how much, true discount? 173. To find the FACE OF A NOTE, when the proceeds, time and rate are given. 6. Required the principal wlien the proceeds are $275.23, le time 93 days and the rate 2% per month. Interest on $1 for 93 days at 2% per months. 062. Proceeds of $1=$1.000 .062=.938. Since there are as many dollars in the principal as the proceeds of $1 is contained times in the proceeds given, S275.230-r-.938 will give the principal required, $293.42+. 176 NELSON'S COMMON-SCHOOL ARITHMETIC. PROOF. Interest on $293.42 for 93 days at 2% per month=18.19-|-, which, subtracted from $293.42., leaves $275.23, the proceeds. 7. The proceeds are $212.60, time 63 days, rate 1J% per numth ; required the principal.* 8. What principal will realize $120 proceeds in 6 months at 10% per annum? 9. The time is three months, rate 10% per an., pro- ceeds $168.97; what is the principal? 10. The rate is 12% per annum, proceeds $693.75, time 4 months; required the principal. 174. To find the RATE PER CENT., when the principal, interest and time are given. 11. The principal is $300, time 60 days, interest $5; required the rate. Interest on $300 for 60 days at 6%=r$3. At 1%=.50. It is obvious that the rate will be as great as the number of times 1% is contained in the interest given. Hence, $5.00-f-50=the rate, 10%. PROOF. Interest on $300 for 60 days at 10% =$5. 12. The principal is $396.15, time 13 months 9 days, interest $26.34,3; required the rate. 13. What is the rate per cent, on $144 for 5 days, when the interest is 24 cents? 14. Eequired the rate on $250 for 60 days, when the; interest is $3.50. 175. To find the TIME, when the principal, rate per cent, and interest are given. Grace being allowed only on notes and drafts, where neither is named, it is not reckoned. *The learner can prove liis work by computing interest on the principal found. GENERAL EXERCISES 177 K-t 15. The principal is $1440, rate 10% per annum, inter- est $37. 50; required the time. Interest on $1440 for 1 day at 10% 40 cents. Since there are as many days as the interest for 1 is contained times in the interest given, $37.50-^40=93^ or 94 days. PROOF. Interest on $1440 for 94 days at 10% per annum=$37.60.* 16. The principal is $1674, rate 2% per month, interest $59.87 ; required the time. 17. In what time will a note for $600, at 6% per an- num, draw $27.50 interest? 1.8. A note for $375 drew $21 interest at 6% per an- num; how long did it require to do it? 19. A merchant wishes to know the time it will take a balance of $917.50 to make $60.80, with interest at 10%. GENERAL EXERCISES. 1. What is the bank discount on a note of $375, drawn at 90 days, at \% per month? 2. What amount of proceeds should I receive from a note of $796, drawn at 60 days, 2% per month? 3. What is the present value of a note drawn for $600 at 30 days? 4. What amount of money should I receive on a note of $675, discounted at 35 days (having 35 days to run), \\% per month? The Teacher can increase these exercises to any extent, as the pupils have to furnish proof of their work; hence, only a few have been given under each artlale. * Interest is never reckoned on the fraction of a day, hence the difference. 178 NELSON'S COMMON-SCHOOL ARITHMETIC. 5. June 3, discounted my note of $350 at 10%, having 30 days to run ; required the discount. 6. February 6, 1858, had A. Seers's note of $500, dated 20th December, 1857, discounted at \\% per month, time to run, 33 days; what were the proceeds? 7. June 3, 1858, discounted my note of December 9, 1857, drawn for $1000, at 12 months, 10%; what had I to pay? 8. October 3, lifted my note of 17th February, drawn for $1000, at 60 days; what amount of money had I to pay? 9. July 30, had M. Norton's note of $360, dated 23d July, at 90 days, discounted at 2% per month, what sum of money (Hd I receive? Answers: $596.72, $11.63, $762.57, $663.19, $1027.50, $2.92, $339.36, $946.67, $491.75. XX. COMPOUND INTEREST. 176. IN Compound Interest, the interest is converted into principal, every quarter, half year or year. Capital is thus more rapidly increased than by simple interest. Any person acquainted with the principles of simple interest will readily understand how to compute this. When there is a settlement of accounts between the parties, after interest has become due and interest is charged in the settlement, interest may be allowed upon the balance found due by the settlement. So an agree- ment, after interest is due, to turn it into principal is valid. Where there is a contract between the parties for the pay- ment of interest annually, if not paid, simple interest may be allowed upon the interest from the time it is due.* *Swan. COMPOUND INTEREST. 179 1. What is the compound interest on $1000 for 2^ years at 6%, payable semi-annually (half-yearly)? The interest of 1000 for 6 mos., $30.00 Add the principal, 1000.00 Amount for 6 mo*., 1030.00 Interest on 1030 for 6 mos., 30.90 Amount for 1 .year, $1060.90 Interest on 1060.90 for 6 mos., 31.827 Amount for 18 mos., $1092.727 Interest on $1092.727 for 6 mos, 32.78181 Amount for Z years, $1125.50881. Interest on $1125.50881 for 6 mos., 33.76526' Amount for 2 years 6 mos., $1159.27407 By deducting the prittopa/, 1000.00 We have the comp\l int. for 2J yrs., $159.27 t 2. What is the compound interest and amount of $672 for 4 years, at 6% per annum? By computing interest on $1 for a number of years, it will be found that the second amount is equal to the square of the first ; the third amount to the cube of the first; the fourth amount to the fourth power of the first; each power corresponding to the number of years. Hence, to find the amount of any principal for any number of years, it is only necessary to multiply the principal by the amount of $1 for the time and rate. Taking the last example, 1.064=the amount for 4 years, which, multiplied by 672= . amount required. 3. Find the amount of $375 for 20 years, at 6% com- pound interest, reckoned annually. At 6 per cent., money will double itself in 11 years, 10 months and 21 days; at 5 per v,eut., in 14 years, 2 months and 15 days; at 8 per cent., in 23 yeai'6, 5 months arid lOJ days. 180 NELSON'S COMMON-SCHOOL ARITHMETIC. 4. Required the compound interest on $600 for 5 years at 1%. Answers: $241.53, $827.68, $848.38, $176.38, $137.50. XXI. AVERAGE. 177. AVERAGE signifies mean or medium. An average number or quantity is one which has an* intermediate value between two or more numbers or quantities. The average between 3, 4 and 5 is 4. 178. The average may be found by dividing the sum of the quantities by their number. 1. Find the average between 3, 4, 5 and 8. 3 EXPLANATION. In these four numbers there are 20 4 parts ; in each of 4 number there are as many equal parts 5 as 4 is contained in 20, viz., 6. _8 4)20 5 2. What is the average price of 5 cows, which cost, re- spectively, $50, $60, $70, $80 and $90? 3. The wages of 9 hands in a factory are $5, $6, $7, $8, $10, $11, $12, $15 and $16, respectively; what is the average ? 4. Required the average length of the following pieces of calico: 37 yds., 35 yds., 31 yds., 30 yds., 32 yds., 27 J yds., 291 ^1^ 3QJ yds., 29| yds. 5. Traveling 5 days, at the rate of 18 miles the first day, 20 the next, 20^ the next, 22J the next and 25 the next, what was the average speed per day? Answers : 70, 3% 10, 21^, 21 T V AVERAGE. 181 EQUATION OF PAYMENTS. 179. When average is applied to the settlement of ac- counts, the process is called Equation of Payments or Equation of Time. Merchants and manufacturers sometimes sell their goods on credit, the time varying from three to nine months, and their customers making numerous purchases before settlement. The object of equation of payments is to ob- tain an average date of payment for these purchases. 6. I owe $3 payable in 2 months, $4 payable in 3 months and 5 payable in 4 months , what will be the av- erage time of payment for the whole amount? The use of $3 for 2 mos^that on $1 for 3x2= 6 mos. The use of $4 for 3'mos=that on $1 for 3x4=12 mos. The use of $5 for 4 mos=that on $1 for 4x5=20 mos. $12 38 mos. Hence, the use of $12 for those several months is equal to that of $1 for 38 months; for 12, it will be -^ as much, or 3^ months or 3 months 5 days. 7. A merchant sells a bill of goods amounting to $4000, to be paid as follows: $400 in 30 days, $600 in 60 days, $1000 in 90 days and the balance in 4 months, or 120 days; what would be a mean or average time of payment for the whole? A credit of $400 for 30 ds. is the same as a credit on $1 for 12000 ds. 600 " 60 " " " " " 1 " 36000 " " 1000 " 90 " " " " " 1 " 90000 " " 2000 " 120 " " " " " 1 " 240000 " 4000 378000 On $1 there is a credit for 378000 days. On $4000, there is a credit for 378000 days divided by ~94| days. 182 NELSON'S COMMON-SCHOOL ARITHMETIC. That is, the $4000 might be paid in 94J days, or on the 95th day, without either party sustaining loss by in- terest. 8. A merchant sells goods to the amount of $1700, $500 payable in 60 days, $300 payable in 90 days and $900 payable in 30 days ; what is the average time of payment of the whole? 9. Sold a bill of goods amounting to $700, J- of which is payable in 90 days, ^ in 4 months and J- in 6 months; required the average time of payment* 180. To find the average date of purchase. 10. Purchased goods as follows; what was the average date of purchase? December 31, a bill of $300; January 3, a bill of $100; January 9, a bill of $200, January 18, a bill of $800; January 23, a bill of $500. 300 X EXPLANATION. The first was due at the 100X 3 300 time of purchase; the second, three days 200 X 9 1800 after; the third, nine days after, etc. 800 X 18=14400 REMARK. If the amounts above were equal, 500X23=11500 an d the intervals also equal, the average date of purchase would be on Jan. 9, because it is 1900 o ' __ midway between the first and last dates. *W Or 15 days after December 31, the date of the first pur- chase, which brings the time up to January 15. If these debts had been contracted on a credit of three months, a note dated January 15 would be given to settle the bill. The time tables (pp. 168, 169) are admirably adapted for averaging, prepared as they are for the two common pe- riods of settlement, January and July. For account sales, the book-keeper should prepare similar ones for each month. *For answers, see end of chapter. AVERAGE. 183 11. The following goods were sold on a credit of 90 days : Required the average date of purchase, or date of note. Jan. 1, Invoice of Coffee 81000.00 Jan. 6, " " Sugar 3500.00 Mar, 9, " " Sunds 9734.00 Mar. 13, " " " 976.50 Apr. 3, " " " 1037.00 16247.50 Required the date of maturity of a 3 months' note, grace included. 12. Sept. 3, Invoice of Calicoes $3150.00 " 19, "Muslins 1174.00 " 20, " " Silks 3500.00 Oct. 19, " " Sundries 1743.00 89567.00 Find the equated time of payment for the following, or date of a sixty-days' note : 13. Apr. 3, 167.25* 14. May 7, $674.40 9, 374.00 Jun. 7, 168.37 " 19, 176.00 . " 10, 370.20 " 20, 371.00 15, 167.00 " 25, 197.87 " 19, 679.60 " 30, 300.00 July 23, 679.45 May 9, 150.57 Aug. 18, 993.18 " 23, 720.18 " 19, 875.57 181. Wlien goods are purchased at different dates and on different lengths of credit. 15. Purchased the following bills of merchandise; re- * When the cents are under 50, reject them; otherwise, add a dol- lar to the dollars. 184 NELSON'S COMMON-SCHOOL ARITHMETIC. quired the average date of maturity, or the equated time of payment for all : Apr. 3, a bill of $250 on 3 months' credit. Apr. 9, " " 157 " 6 " " May 7, " " 250 " 4 " " Jun. 9, " 320 " 2 " " If we substitute the date of maturity of each of these bills for the date of purchase, and arrange them in the order of time, we shall have a problem in all respects sim- ilar to those under last Art. The first bill falls due July 3;* the second, October 9; the third, September 7; and the fourth, August 9. Ar- ranged in the order of time, they appear thus : July 3, $250 X 2 500 By the July table we find the Aug. 9, 320 X 3912480 difference of time to be 2, 39, 68 Sep. 7, $250 X 6817000 and 100 from the first, and the av- Oct. 9, $157 X 10d 15700 era e time > 46 an(i over a half, or 47 **?*' 47 da * s after 977 )45680(46 3908 August 17. 6600 5862 "738 When time tables are used, especially such as are adopted for periodical settlements, as those in banking, the purchases need not be arranged in the order of date, as above. 16. Find the average date of payment for the following: Feb. 3, Mdse, 3 mos, $678.59 July 27, Mdse, 30 ds, $1500.00 Mar. 9, 2 " 243.75 Aug. 9, " 90 " 175.50 Apr. 13, " 4 1000.00 Oct. 3, 2 mos, 1673.13 Jun. 17, 30 ds, 976.54 Nov. 18, " 3 " 987.65 19, 3 mos, 786.15 Dec. 13, 3 685.18 *Days of grace are not allowed on invoices. AVERAGE. 185 182. 'When cash goods are sold with others on credit. Groods are sometimes classed as cash or time. Those designated cash do not always realize present payment, any time within a month being considered cash; and even at the end of a month cash has not been exacted, the un- derstandino- being that a debt contracted on cash terms o o draws interest from date. When computing average, cash bills are considered due on the day of purchase. 17 Jan. 1, Feb. 6, Mar. 8, Apr. 17, 900 X 59=r 53100 800X125=100000 700 X 66= 46200 600X106= 63600 3000 JOHN MICKLEBOROUGH. To Mdse on 2 months $900.flD " " "3 " . . 800.00 " " " cash 700.00 " " " 600.00 EXPLANATION. 2 months from Janu- ary 1 is March 1, or 59 days; 3 months from February 6 is May 6, or 125 days from January 1 ; March 8 is 66 days from January 1, and April 17 is 106 days from January 1. Hence, March 30 is the aver- age date of maturity or payment. 88 days after Jan. 1 or Mar. 30. )262900 Find the average date of maturity of the following: D. E. SOMERSET. A. E. NELSON. 18 Jan. 1, On 3 mos, $600.00 Feb. 3, For cash, 670.00 Mar. 3, 6n 6 mos, 950.00 May 3, For cash, 550.00 July 1, On 3 mos, 13, On 2 mos, 19, For cash, 23, On 5 mos, 19 $675.00 619.54 147.67 678.44 20 Sept. 3, At 30 days, $937.15 9, At 90 days, 897.78 17, Cash, 619.18 Oct. 3, At 60 days, 777.00 16 186 NELSON'S COMMON-SCHOOL ARITHMETIC. Required the amount due on each of the following on July 1 : 21. Jan. 9, $678.44 at 60 days; 20th, $419.88 at cash price; 29th, $789.14 at 3 mos. 22. April 9, $1678 on 3 mos; June 18, $1000 at cash price; 21st, $879.55 on 60 days; 23d, $371.19 cash; 29th, $785.25 cash. 183. Average Applied to statements when there are credits. Dr. C. A. WALWORTH. Or. 1867. 1867. Jan. 1, To Mdse, $300.00 Feb. 9, By Mdse, $200.00 Mar. 3, " 500.00 23, u " 100.00 300X 200X39=7800 500=61=30500 Dr. products. 100x53=5300 gjJo 13100 Or. products. ^ ^^ 300 500)17400 Bal. due 500 34|- or 35 days from Jan. 1 or Feb. 5. EXPLANATION. Assuming both purchases and sales to be due on January 1, W. would be entitled to a discount on his purchases equal to that on $1 for 30500 days, and I would be entitled to a discount on my purchases equal to that on $1 for 13100 days, mak- ing a difference of 17400 days in W.'s favor on the balance, $500. The discount on $1 for 17400 days is equal to the discount on $500 for 17400 days^-500r=34| days from January 1. Whether to bo reckoned backward or forward is easily determined. To be in his favor, the time for payment must be reckoned forward; otherwise, it would be reckoned backward. Required the equated time of the following: WILLIAM SILLETS. 24 Feb. 3, To Mdse, $250.00 Mar. 2, By Cash, $300.00 Mar. 9, " " 300.00 Apr. 3, " " 200.00 Apr. 18, 500.00 AVERAGE. 187 Dr. J. C. HINTZ. Or. 1869. 1869. July 3, To Mdse, $1000.00 Aug. 1, Bj Cash, $500.00 7, " " 500.00 13, " 500.00 Au. 18, " " 250.00 Assuming the day of settlement to be July 1, we have 1000 X 2= 2000 Days. 500x31=15500 Days. 500X 6= 3000 " 500x43=21500 " J60 X 48=12000 i^T 3WM Or. products. 1750 17000 " 17000 Dr. products. 1000 20000 Difference of do. $750 Balance due. 750)20000(26 ds. EXPLANATION. The sum of the credit pro- 150 ducts being greater than that of the debit ~ products, shows that the discount is in my favor. Hence, in order to settle on the assumed 4'~){) J date, his payments would be at a discount of 50 26 days for the balance; but as it would be impossible to settle on a past date, we will have to charge him interest from 20 days prior to July 1 (June 5) to the real day of settlement, whatever that may be, or else take his note, dated June 6, bearing interest from date. Say the date of the settlement is September 1. Interest on $750 from June 5 to September 1 is $11, which, added to $750 $761, balance due. This balance is 9 cents less than what would be obtained by computing interest on both sides of the account to Sep- tember 1, and is caused by ignoring the T ^ of a day 26 days being taken instead of 26^. NOTE. Book-keepers sometimes omit the tens of dollars when averaging. In the above example the hundreds might have been omitted without serious error. 188 NELSON'S COMMON-SCHOOL ARITHMETIC. Dr. WILLIAM P. WALLACE. Or. 1869. 1869. Feb. 22, To Mdse. $500.00 Apr. 3, By Cash, $620.00 Mar. 28, " " 700.00 July 6, " " 520.00 Apr. 30, " " 900.00 Sep. 10, " " 900.00 Juii. 8, " " 600.00 Dec. 1, 650.00 Assuming January 1 as the day of settlement, we have the following formula: 500X 52= 26000 620X 92= 57040 700X 86= 60200 520x186= 96720 900x119=107100 900x252=226800 600X158= 94800 650x334=217100 2700 288100 2690 597660 2690 288100 10 10)309560 30956 days, Or 84 years, 9 months and 5 days (allowing for leap-years} to be counted backward, because the discount is in my * j favor. 184. Hence, when the balance of the account and tho balance of the discount are both in my favor, I coun backward; when the former only, I count forward, ano vice versa. 29. Goods bought on January 9, at 60 days, $1376; on 13th, $780; on May 3, $3400. Payments made May 3 $1200; June 8, $3500; equated time required. 30. Balance of last account brought down, including in: terest to July 1 ; goods bought August 9, at 3 months $2300; September 3, $1500; December 9, $500; and pay ments, October 3, $3000; November 9, $2500; averag. date of payment required. 31 to 33. Find the equated time of the following : AVERAGE. 189 1869. July 3, To Mdse, 6 mos, 15, " " 3 " u 3 tt : < Cash, H. H. SCHULTZ. 1869. An. 21, Sep. 18, Oct. 15, 21, 27, 31, Nov. 28, 30, Cash, Mdse, 3 mos, u 4 u 560.87 149.50 2000.00 396.40 175.20 425.16 100.00 506.18 197.45 321.16 July 1, By Balance, 127.15 30, " Accept. 60 ds, 300.00 Aug. 29, Cash, 460.00 Oct. 20, " Note, 3 mos, 1000.00 31, 31, Nov. 30, 30, Cash Mdse Ret., Cash, Balance, 100.00 250.00 450.00 2144.77 4831.92 Dec. 1, 1867. To Bal., 2144.77 THEODORE LILIENTHAL. 1867. Jan. t, To Balance, 650.00 Jan. 8, By Mdse, 3 mos 160.00 Feb. 3, u Cash, 245.00 15, (t it 6 " 710.87 15, it Note, 60 ds 416.87 Feb. H u u 2 910.14 Mar.18, it Accept, 30 ds, 1000.00 Apr. 16, It 11 Cash, 1000.00 Jun. 4, it a 60 " 750.14 June 8, 11 It 4 mos, 900.00 16, it Note, 3 mos, 987.64 15, tl tl 4 2500.00 30, tt Cash, 500.00 17, tt It 6 " 1215.00 30, tt Mdse, abate. 200.00 30, tl Sunds i 700.00 JULIUS WISE. 1869. 1869. July 3, To Balance, 1500.00 Aug. 3, By Cash, 1000.00 18, tt Mdse , 4 mos, 750.40 Sept. 7, U Acctc . 60 ds, 500.00 Aug. 29, it it 4 128.80 Nov. 5, 11 it 60 " 750.00 Sep. 30, tt tt 4 " 916.84 Dec. 14, tt Cash, 2000.00 Oct. 10, u it 3 500.00 30, " Cash, 675.14 Nov. 18, " Sunds, 564.18 These exercises may be omitted until the learner has studied Exchange. 188. When payments are made before a note or bill is due, to find how long after maturity it should run to bal- ance the interest on the advanced payments. 190 NELSON'S COMMON-SCHOOL ARITHMETIC. 34. A merchant holds a note of $500 at 6 months. Three months before it is due, he receives $100, and one month before it is due, he receives $300 ; how long should he allow the balance to run to equal the interest on the advance? The int. on 100 for 3 mos=int. on $1 for 300 mos. The int. on 300 for 1 mo r^int. on 1 for 300 mos. 600 mos. Hence, the interest on the advance payments is equal to the interest of $1 for GOO mouths ; that is, a balance of $1 should have run 600 months, but the balance due on the note is 100; therefore, it should run |-{^ months 6 mouths. PROOF. The int. of the $100 for 6 mos=$ 3. The int. of $100 (the first pay't) for the 3 mos=$1.50 The int. of 300 (the sec. pay't) for the 1 mo = 1.50 Total interest on advance=$3.00 35. A note of $600 was given January 3, 1868, payable in 6 months. 4 months before it was due, $100 was p;iid on it, and 3 mouths before it was due, $200 was paid; how long in equity should the balance run? 36. A merchant owes $700, due 8 months from the time he contracted the bill ; 5 months afterward, he pays $200, and two months after that, $300; how long should the. balance remain unpaid? 37. If I borrow $600 from A at one time, and $500 at another, each for 4 months, how long should I lend him $1000 to return the favor? 38. I owe $400, payable in 10 months; at the end of 4 months I pay $100; 3 months after that, $50; how long after the expiration of 10 months may the balance remain unpaid ? AVERAGE. 191 39. A owes $1000 due in 6 months; 5 months before it is due he pays $200, and 3 months before it is due he pays 300 ; how long after the expiration of the 6 months may the balance remain unpaid? 187. Average applied to account sales. An account sales is a detailed statement of goods re- ceived by a commission merchant, and sold on account of another. The person who sends the goods is called the shipper or consignor; the person to whom they are sent, the consignee, and the goods, the consignment. The duty of an agent or commission merchant is to pro- cure the best intelligence of the state of trade at the place where he does business, including the quality and quan- tity of goods in the market, their present prices, and the probability of their rising or falling; to pay exact obedi- ence to the orders of his employers; to consult their ad- vantage in matters left to his discretion; to execute their business with all the dispatch circumstances will admit; to be early in his intelligence, distinct and correct in his accounts, and punctual in his correspondence. 188. An account sales should state from whom the goods were received, or on whose account and risk they were sold, the dates and terms of sales, the cash paid for freight, drayage, etc., and the various charges, such as in- i suraiice, commission, cooperage, etc. The total amount received will appear on the right of the account, and the charges, etc., on the left, or they may be arranged as in the example. The difference between the two sides is called the pro- ceeds. Advances made on goods are charged to the shipper's account, not in the account sales. Where goods are sold promptly for cash, or on short time, the account sales is not averaged. 192 NELSON'S COMMON-SCHOOL ARITHMETIC. COMMISSION HOUSE OF STRAIGHT, DEMING & CO. Shipment 18. No. 7828. Sales for account of Messrs. Gaff & Baldwin. By sundries, June 4, T. B. Colgan & Co. @ 60 days, 8 hhds. Sugar. 1095 1020 1100 1120 1080 1240 8965 1200 1110 896 8069 @ ?& $600.13 June 6, G. Newton & Co., @ 60 days, 10 hhds. Sugar. 1080 1040 1090 1340 1120 1020 1240 1100 11440 1200 1210 1144 10296 @ 6| 8707.85 June 10, B. Vilgers & Co., @ 60 days, 20 hbds Sugar. 1060 1240 1210 1110 1180 1005 1055 1285 1240 1100 1185 1210 1300 1325 1010 1140 1120 1205 23185 1205 1000 2318 20867 6^ 1343.31 $2651.29 CHARGES. June 1, P'd cash st'r Landis for freight, $87.18 10, Dray. 9 50 , ins. 4 63 , and stor. 9 50 , 23.63 " Commission and guarantee, 132.56 243.37 Net proceeds due by equation, Aug. 13, $2407.92 E. 0. E. CINCINNATI, June 14, 1858. STRAIGHT, DEMING & Co., Per F. JELKE. AVERAGE. 193 Aug. 3, 600X63^37800 June 1, 87 " 5, 708X65= 46020 " 10, 156X9=1404 9, 1343X69= 92667 ^ ^ 2651 )176487 243 1404 2408 175083(72.6 or 73 days from June 1, 16856 which gives August 13. 6523 4816 1707 2. Find the equated time of payment of the following: Account sales of merchandise sold on account and risk of Morris J. Parry, New York: Mar. 1, To Cash for freight, $50.00 Mar. 15, By Cash, $450.00 1, " Drayuge, 10.00 Apr. 3, " S. Miner, 75.00 10, " Insurance, 8.50 May 1, " Cash, 31rf.75 June 20, " Storage* Advertising, 10.00 7, "J.Clark. 92.25 " Com. on $1224 @ 2,^, 30.60 Jun. 19, " Cash, 288.00 M. J. Parry's net pro., 1114.90 $1224.00 $1224.00 3. Sales of 100 bbls. of molasses for acct. of C. H. Crane. July 3, F. M. Peale, on acct., 20 bbls., 860 gals... @ 1.25 1075.00 July 9, Saml. A. Butts, Jr., cash, 15 bbls., 635 gals. @, 1.20 762.00 July 18, Geo. T. Ladd, cash, 12 bbls., 495 gals @ 1.20 594.00 Sep. 6, F. M. Peale, on acct., 12 bbls., 495 gals... @ 1.20 594.00 Oct. 3, J. J. Marvin, on acct., 31 bbls., 1285 gals. @ 1.00 1265.00 Dec. 18, J. J. Marvin, on acct., 10 bbls., 400 gals.. @ .90 360.00 $4650.00 CHARGES. July 1, Cash paid freight 58.00 Drayage 18.00 Dec. 30, Cooperage 3.20 Com. and guar. 4 percent 186.00 C. H. Crane's net proceeds 4384.80 $4650.00 17 194 NELSON'S COMMON-SCHOOL ARITHMETIC. SALES FOR ACCT. OF R. H. LANGDALE. Sept. 3, G. F. Sands, 90 days, 18 bbls. Clo. S. 3120 Ibs net=52 bus. @ $6.50 338.00 Oct. 9, C. F. Howe, on 60 ds. note, 5 bbls. Clo. S. 930 Ibs., 15J bus... @ 10.00 155.00 Dec. 5, G. A. Voige, 30 ds. note, 10 bbls, 2220 37 @ 8.00 296.00 789.00 CHARGES. Aug. 15, Cash pd. fr't 12.00 Drayage on acct 6.00 Sept. 1, Advertising 8.55 Dec. 30, Storage 10.00 Com. and guar., 5 per cent 39.45 R. H. Langdale's net proceeds 713.00 799.00 E. E. CINCINNATI, Jan. 1, 1867. NELSON, NEPHEW Co. Answers: 3 mos. 24 ds.; 3 mos. 10 ds. ; 3 mos. ; Deo. 31, 1869; 4 mos. 15 ds.; July 15, 1869; Dec. 15, 1867; $1919.85; 4 mos. 12 ds. ; Oct. 17; March 25; $4707.34; Nov. 2; 49 ds. ; Feb. 21; July 11; April 30; Dec. 23; 143 ds. ; April 27 ; Dec. 6 ; Oct. 8 ; May 17 ; Dec. 21 ; Aug. 24; June 2; Aug. 16, 1867. ACCOUNTS CURRENT. An Account Current is a statement of the entire trans- actions between two parties, generally for three or six months. It exhibits the whole sums given and received, the interest due on each at the date of the account, and to whom the balance of interest an9 principal is due. Without a knowledge of book-keeping, an account cur- rent or account sales can not be fully understood. The name of the party against whom the account cur- rent is made out is always written first and on the left, while the maker's name appears on the right. When any sums fall due after the date of the account, the interest ia entered on the opposite side, making it discount. AVERAGE. 195 AN ACCOUNT CURRENT WITH INTEREST COMPUTED. MESSRS. GRAFF & BALDWIN, In Account Current and Interest Account with Dr. STRAIGHT, DEMING & Co. DATE. DKSORIPTION. TIME. IN'T. AMOUNT. 1867. Jan. tt Feb. Mar. Apr. 4 10 8 20 2 1.0 18 To Mdse, as per bill rendered Days 161 155 63 114 104 96 57 55 44 49 8 37 20 9 6 145 137 88 77 65 37 16 11 20 19 2 4 4 1 8 4 c. 40 93 25 00 17 08 12 02. 10 i 99 67i 03 92 74 15 17! 425 810 500 1000 125 5 434 504 150 1101 500 5 275 495 4152 5 30 c. 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 03 tt It U (( " Your draft on us at 60 days sight, in favor of II King & Co " Your sight draft on us (t Mdse, as per bill rendered . ... " Exchange on $500 at 1 per cent.... " Mdse, as per bill rendered tt May June u Jan. Mar. tt Apr. May u " June 20 25 26 4 8 25 5 8 14 20 28 18 29 10 8 29 14 " Our draft on G. Wright, your favor. " N. Davis & Co.'s note at 30 days, tf Discount on uncurrent money at 1/1* P er cent l< Mdse, as per bill rendered " Net proceeds, sales 100 bbls. flour. " Mdse, as per bill rendered " Cash paid for vour telegram " Interest in our favor CR. By draft on !New York, % P- ? prem. " Cash 84 24 2 11 5 6 1 2 30 84 10515 03 29 28 74 17 76 23 67 03 17 1005 100 800 403 624 200 1000 2407 3976 10515 00 00 00 00 00 00 00 00 03 " Your sight draft on Moore & Adams, our favor " Draft on New York, % P er cent. " Net proceeds, sales 50 bbls. ino- Cash " Net proceeds, sales 38 hhds. sugar. E & E CINCINNATI, June 14, 1867. STRAIGHT, DEMING & Co., per HILL. 03 196 NELSON'S COMMON-SCHOOL ARITHMETIC. XXII. EXCHANGE. 189. IF A of Liverpool is indebted to B of New York in the sum of $1000 dollars, and B is indebted to C of Liverpool to the same amount, it is evident that the two debts might be paid without either party sending real money to the other. A settlement could be effected by B sending the following order to C : $1000. NEW YORK, Jan. 3, 1868. At sight, pay to the order of C One thousand dol- lars, value received, and charge to my account. To A, B Liverpool, England. This method of settling accounts between persons in distant places is called Exchange. If between persons in the same country, it is called Home, Domestic or Inland Exchange; otherwise, it is called Foreign Exchange. 190. A Bill of Exchange is an order written like the one above ; it is also called a check or draft. The term draft is usually applied to inland bills. There are Sight Bills and Time Bills. 191. A Sight Bill is one payable at sight or on de- mand, as the one above. A Time Bill is one that requires payment so many days after sight, or after date. 192. A Set of Exchange consists of three copies of the same bill drawn to insure safety of transmission, one of which being paid, the others are void. The person whose signature is attached to a bill or draft is called the drawer or maker; the person addressed, the drawee; the party to whom payment is to be made, EXCHANGE. 197 the payee, and the one who has possession of it ? the owner or holder. REMARKS ON BILLS AND DRAFTS. 1. Bills of exchange, unless payable at sight or on de- mand, require to be " accepted," and should be presented promptly to the drawee for this purpose. The drawer then becomes the acceptor, and the bill is said to be ac- cepted or honored, and is called an acceptance. 2. Accepting a bill consists in the drawee writing his name across the face, by which act he becomes responsible for its payment. The following is the form: &C CO. CO Business men prefer a draft to a promissory note, be- cause there are three parties to it, while the note, ordi- narily, has only two. It often happens, however, that notes and drafts are made payable to the drawers, which leaves only one party to the former, and two to the lat- ter, though technically they are considered the same as if drawn in favor of others. When bankers receive unaccepted bills, they send them out for acceptance or notify the drawees. 198 NELSON'S COMMON-SCHOOL ARITHMETIC Bills of exchange, like promissory notes, may be made negotiable by the insertion of the words, " to the order of," or "bearer," and are subject to protest for non-pay- ment, and the indorsement may be special or in blank. 3. Bills of exchange can be had at the banking houses or offices of exchange brokers, and may be drawn in favor of the persons buying them, indorsed and mailed to their creditors; or they may be drawn in favor of some other person, indorsed by him, and afterward indorsed by the buyer, and mailed to his creditors, as before. 4. When a bill or draft costs neither more nor less than the amount of its face, it is said to be at par; if less than that amount, it is at a discount; if more, it is said to com- mand a premium, and the rate of discount or premium is called the rate of exchange. 5. The phrase, "apply to my account," or "your ac- count," "as advised," etc., are not essential to a bill, but rather indicate the relations of the parties as debtors or creditors. When the drawer is indebted to the drawee, he would say, " apply to my account," but when the drawee is indebted to the drawer, the phrase would be, " apply to your account," or "put it to your account." Should the bill be drawn on account of a third person, he would say, "put it to the account of A." 6. When the words "as per advice," or "as advised," are used, it is presumed that a letter of instructions has preceded the draft. In such case, the drawee honors at his risk in the absence of such advice. 7. Bills or drafts for acceptance must be presented within a reasonable time. If the drawee destroy a bill for acceptance, or refuse to return it in twenty-four hours, he be deemed to have accepted it. EXCHANGE. 199 8. Sight bills for collection should not be mailed to the drawee, as their possession is presumptive evidence of payment. 9. The phrase "value received" is properly omitted when the bill is drawn against funds of the drawer in the hands of the drawee, as is usually the case with banking houses when selling exchange. 10. The place of payment, separate from where it is drawn, is not usually inserted in a draft, unless an under- standing to the contrary exists between the parties. 11. Drafts are often drawn by merchants upon each other to raise money or settle accounts. A merchant shipping a large quantity of goods to another to sell on commission, usually draws a draft for a part of the cost on the party and sells it at bank, or passes it to another merchant in the course of business. This kind of paper is called a mercantile draft, to distinguish it from one is- sued by a bank, which is called exchange, or a bank check or draft, and is not so available for transmission as the bank draft or exchange. It is a part of the business of a banking house or exchange office to buy this mercantile paper, send it home for collection, and in the mean time sell exchange on the banks to which they transmit it, for such sums as may be demanded. DRAFT OF A MERCHANT UPON ANOTHER TO WHOM HE HAS SHIPPED GOODS. NEW YORK, May 17, 1867. , At ten days' sight, pay to our order One thousand dol- lars, value received, and charge to our account. To HENRY L. WEHMER. A. J. RICKOFF & Co. Cincinnati, Ohio. To obtain money on this, Mr. Rickoff would indorse it to a banking house, which would pay him the current 200 NELSON'S COMMON-SCHOOL ARITHMETIC. rate for mercantile paper, and discount for the time to ex- pire before collection say 4 days to reach its destination, and 13 days for maturity. BILLS IN DUPLICATE AND SETS OF EXCHANGE. To prevent delay and guard against loss, bills of ex- change are often drawn in duplicate or in sets.* A duplicate bill is, as the name indicates, a correct copy of the original, with the addition of the word " Dupli- cate" written or printed across the face.f A set of exchange properly belongs to Foreign Ex- change, under which subject the learner will find the form and description. HOME EXERCISES IN DRAWING DRAFTS. 1. Draw on your teacher, (locating him in San Fran- cisco, California,) at 10 days' sight, for $2136.50, and make the draft payable to yourself. 2. At 60 days after date, draw on 0. I. Mitchell for $2000, favor of yourself, and prepare draft for negotiation. Draw bills and notes from the following data: 3. Draw on M. Garaghan, St. Louis, for $3600, at sight, and prepare the draft for collection by the Central Na- tional Bank, Cincinnati. * Should a bill be lost in transmission, the amount can be recov- ered of the bank from which it was bought, unless it can be proved that payment was made by the drawee. t On bank drafts will often be found writing on both back and face, which can not be represented in type, such as the names of bank officers, through whose hands, they pass before issue, the amount written on the back or face a second time to prevent alter- ation. They are also drawn payable in "gold" or "currency," as occasion requires. EXCHANGE. 201 Drawer, Drawee, Payee, Date, Where, Time, Amount, 4. 5. Yourself. Drawer, Yourself. Your teacher, at N. 0. Drawee, W. A.Fillmore,N.Y. G. A. Carnahan. Date, The present. Time, Cincinnati, Ohio. Payee, April 1, 1867. 10 days' sight. Yourself. At sight. $367.25. Amount, $5907.84. 6. Face, $3167.85. Date, August 9, 1866. Payee, B. 0. M. De Beck. Indorsee, Luther W. Strafer. Drawer, Noble K. Royse, Cincinnati, Ohio. Drawee, Herman H. Raschig, New Orleans. Time, 10 days' sight. 7. ACCEPTANCE OF F. M. PEALE AT NEW YORK. Drawer, Wm. H. Morgan, San Francisco. 1st Indorser, J. M. Allen. Time, 60 days after date. Face, $967.18. Date, 24th May, 1866. Indorsee, Geo. F. Sands. 8. Drawer, John J. Marvin, Cincinnati, Ohio. Drawee, John S. Highlands, Columbus, Ohio. Payee, E. H. Prichard, Boston, Mass. 1st Indorser, B. B. Stewart, Face, $2127. Time, 30 days after sight. Date, January 1, 1867. Accepted two days afterward. 9. Indorsee, G. W. Harper. Drawee, C. R. Stuntz, Washington, D. C. Drawer, G. W. Smith, New York. Indorser, G. A. Schmitt. Time, 10 days' sight, Face, $5000. Date, January 1, 1867. Accepted four days after date. 202 NELSON'S COMMON-SCHOOL ARITHMETIC. AN INDIANA NOTE. $500. INDIANAPOLIS, May 25, 1867. Six months after date, I, the subscriber, of Indianapolis, County of Marion, State of Indiana, promise to pay to the order of Geo. W. Runyan, Five hundred dollars, without any relief whatever from valuation or appraisement laws. Value received. Payable at First National Bank. JAMES W. WILSON. No. 59. Due November 28. 1867 XXIII. FRACTION'S. 193. A FRACTION is a part of one or more things con- sidered as a whole, and is therefore the result of division. Whole numbers are sometimes expressed in fractional form. (See Improper Fractions.) 194. The subject of fractions is the method of treating fractional numbers, or showing how they may be added, subtracted, multiplied and divided. 195. Fractions are divided into common and decimal, according to the way in which they are written. A com- mon fraction requires two numbers to express it, as J, while a decimal requires only one, with a period at the left, as .5. The following fractions would be read as shown oppo- site: jiy One Twelfth. fr Three Seventeenths. Nine One hundred and forty -fifths. Fifty seven Three thousand three hundred and ninety -sixths, or Thirty-three hundred, etc. FRACTIONS. 203 198. The two numbers forming a fraction are called terms; the upper term the numerator, and the lower term the denominator. The line between the terms is the sign of division, and indicates that the upper term is divided by the lower. (Art. 34.) -f% represents the twelfth part of 3, or 3 parts of something divided into 12 parts. A fraction also expresses the ratio between the two terms. ^=3 : 12. PRINCIPLES OF FRACTIONS. 197. If both terms of a fraction be multiplied by the same number, the value of the fraction will remain un- altered. Let the 1 and 4 of the fraction J be multiplied by 2, and we have -|, a fraction of the same value as J. 198. If both terms of a fraction be divided by the same number, the value of the fraction will remain un- altered. Let 3 and 6 of the fraction - be divided by 3, and we have ^, a fraction of the same value as -|. 199. If the numerator only be multiplied, the value of the fraction will be increased and the whole fraction mul- tiplied. Let 2 of the fraction f be multiplied by 4, and we have |; that is, eight thirds instead of two thirds. 200. If the denominator only be multiplied, the value of the fraction will be decreased, and the whole fraction divided. Let the 3 of the fraction be multiplied by 2, and we have | ; that is, two sixths instead of two thirds. 201. If the numerator only be divided, the value of the fraction will be decreased, and the whole fraction divided. Let 4 of the fraction |- be divided by 2, and we have |-; that is, two eighths instead of four eighths. 204 NELSON'S COMMON-SCHOOL ARITHMETIC. 202. If the denominator only be divided, the value of the fraction will be increased, and the whole fraction mul- tiplied. Let 8 of the fraction -| be divided by 2, and we have J ; that is, two fourths instead of two eighths. Common fractions are divided into simple, proper, im- proper, contfpvund, complex fractions and mixed numbers. A simple fraction is a single fraction, as f . A proper fraction is a single fraction whose numerator is less than the denominator, as jr. An improper fraction is a single fraction whose numer- ator is equal to or greater than the denominator, as f, f , which indicates not a part, but the whole or more than the whole; hence, the term improper. 203. A compound fraction is a fraction of a fraction or fractions, as % of f or 4 of -J-* of ||. 204. A complex fraction is one having a fraction in the numerator or denominator, or in both, as 205. A mixed number is composed of a fraction and whole number together, as 7f . REDUCTION OF FRACTIONS 206. Fractions are often expressed in terms too large for convenient use, or to estimate their value at sight. The fraction -^ possesses the same value as J, and for convenience in operating ought to be reduced to that de- gree of simplicity. The process of changing the form of a fraction in this manner is called reducing it. 207. To reduce a fraction to its lowest terms. FRACTIONS. 205 We divide both terms by any number or numbers which will do so without a remainder. (Art. 198.) 1. 9 and 27 of the fraction ^ divided by 9, give 1 and 3, or the fraction ^. When a single number will not reduce a fraction to its lowest terms, other numbers are used and the process con- tinued.* 2. To reduce -- to its lowest terms. < 3 to 13. Reduce the following fractions to their lowest terms: H> 4tt. TS, iWr. T#&. T&V&. dMr. isWff. If!!. 9 3 6 9876 2"8lT8> TSTTOT- Answers : |, |, yffj-, ^fff , |, 3 \, |, f f , #& i dhr- 208. ^ raise a fraction to a higher denomination, we multiply both terms by the same number a process the reverse of the last. 209. When the higher denomination is given, the mul- tiplier may be obtained by dividing the new denominator by the old. *THE GREATEST COMMON DIVISOR Is the greatest number which will divide any two or more numbers. It may be found by the following process, but the operation is so long that it is seldom used in practice. To find the greatest common divisor of 540 and 612. 640)612(1 EXPLANATION. The smaller number is di- 540 vided into the larger, and the remainder (72) 72)540(7 into the first divisor; then the next remainder 504 (36) into the last divisor, etc. 36)72(2 The last divisor is the greatest common di- 7^ visor, viz., 36. That is, no number higher than 36 will divide both without a remainder. 206 NELSON'S COMMON-SCHOOL ARITHMETIC. 14. To raise ^ to 24ths, we divide 24 by 6, which gives 4 as a multiplier. 5x420 an d 6x4=24, making |f. 15 to 21. Raise f to 12ths, $ to IGths, -f to 49ths, | to 28ths, | to 120ths, -^ to ISOths, ff to 500ths. 210. To reduce a mixed or whole number to an improper fraction. 22. Reduce 5|- to an improper fraction ; that is, in 5-J- how many eighths? 5? EXPLANATION. In every whole number there are 8 eighths, 8 and in 5 whole numbers there are 8 times 5, or 40 eighths, "TT to which add 7 eighths and the result is 47 eighths. -r 23 to 30. Reduce the following numbers to improper fractions: 15f, 7f 6$, 51f, 17, 113ft, U, 21 f Answers : -*f<s fy M, ^, *f> Ai f> S ^^ 211. ^o reduce improper fractions to whole or mixed numbers, is an operation the reverse of the last. 31. Reduce ^ to a mixed number. 8)47 ~&i 32 to 41. Reduce the following to whole or mixed num- bers, and the remaining fractions to their lowest terms: "t 4 . H*. -f A > W. Hf 6 -> - x r. -H 6 - -. ^iW, WA H?- -- Answers: 182, 10ft, 2% 50^, 12^ 693^, 17#V. XXIV. DECIMALS. 212. A DECIMAL FRACTION expresses its value in one term, and is known from a whole number by its having a period, called a decimal point, at the left. .5 is a decimal. The value of a decimal is more easily ascertained than DECIMALS. 207 that of a common fraction, while operations in decimals are performed with nearly the same ease as those in whole numbers. 213. Figures increase in a tenfold ratio as they are re- moved one place to the left, and decrease in the same ratio as they are removed one place to the right. In the number .5 the figure is one place to the right of the unit figure, and therefore possesses only one-tenth of the value it would in that place. In other words, it represents tenths instead of units. One place further it would represent hundredths, as .05, and one place further, thousandths, as .005. As common fractions these would appear thus : ^, yjfo T f^ ; hence, 214. To reduce a decimal to a common fraction, we erase the decimal point, and write for the denominator as many ciphers as there are figures in the numerator, and prefix the figure 1. .075 would be written yj-g-g- and 0047, TD Vw- Ciphers on the extreme right of a decimal possess no value. .500 expresses the same value as .5, the first be- ing -j^nnr, which, reduced to tenths, is 1 to 21. Find the fractional value of the following and read them: .23, .007, .013, .760, .00019, .3401, .67800, .0907, .0076, .3467, .1093, .0770, .3657, .2136, .09876, .000001, .13607, .06789, .03146, .000016, .016037. 215. As in Federal money, the removal of the decimaJ point one place to the right multiplies a number by 10, and its removal one place to the left divides it by 10. Removed to the right in 1673.27, we have 16732.7, and removed to the left we have 167.327. The notation and numeration of decimals being so sim- ilar to that of whole numbers, little trouble will be expe- rienced in reading the following: 208 NELSON'S COMMON-SCHOOL ARITHMETIC. 22. .01 28. .3013 34. .45689 40. .00001 23. .57 29. .0031 35. .18654 41. .0010906 24. .709 30. .2160 36. .36109 42. .3016031 25. .856 31. .1061 37. .00009 43. .0016039 26. .2913 32. .4064 38. .01002 44. .0000067 27. .0016 33. .5067 39. .168002 45. .00001001 Express in figures the following: 1. One tenth. - 2. Three hundredths. 3. Five thousandths. 4. Sixty-five ten thousandths. 5. Three hundred seventy-six thousandths. 6. Four hundred ten thousandths. 7. Eighteen hundred and twenty thousandths. 8. Sixty-one fand eighteen hundredths. 9. Forty-five hundredths. 10. Eighty-seven thousand jand sixty hundredths. 11. Five hundred thousand and seven tenths. 12. One hundred and one thousand and seven. 13. Sixty-four thousand and eight tenths. 14. Nine millions. and seventy-nine thousandths. 15. Eighty-six hundredths. 16. Seven thousand and six hundredths. 17. One hundred and ten and sixty-five hundredths. 18. Eighteen hundred and sixty-seven^and seventy-five hundredths. 19. Twenty-four hundred and five hundred and one thousandths. ADDITION OF DECIMALS. 216. When arranged, tenths under tenths, hundredths under hundredths, etc., decimals are added and subtracted precisely as whole numbers. The operations in Federal DECIMALS. 209 money, with which the learner is already familiar, properly belong to this subject. In those, the decimal points were placed directly under each other. The same rule should be observed in adding or subtracting decimals generally. I. To add 1.07+.001+37.045+10.06+.0007. 1.07 EXPLANATION. Here the decimal points are arranged .001 directly under each other and addition performed as in 37.045 whole numbers. 10.06 .0007 48.1767 2. 2.13 + .426 + 21.2 + 7.63 + 640.072=? 3. 43.27 + 9.042 +712.417 + 41.007 + .962=? 4. 820.71 + 2.006 + 84.243 +217.072 + 9.341=? 5. 107.67 + 1.301 + 20.0163+684.6 + 10.06 =? 6. 719.86 + .2103+ .1610+310.6 +2134. =? 7. 9.8784+29.8 + 67.19 + 7.916 + 379. =? 8. 643.72 + .109 +360.06 + .0006+ .216=? Answers: 2748.3678, 671.458, 1133.372, 823.6473, 806.698, 1004.1056, 493.7844, 2748.3678, 2708.3768, 493.7844, 3164.8313. 9. .007+31.06 + .1009+100.07 =? 10. 710.34 + 2.406 +67.709 + .0006=? II. 314.60 + .0006+ .0027+ .001 =? 12. 714.06 + .003 + 8.007 +800 =? Answers: 314.6043, 1522.07, 131.2379, 780.4556, 1522.3074. SUBTRACTION OF DECIMALS. When the larger number has fewer places of decimals than the smaller, the blanks may be filled with ciphers. (Art. 214.) 18 210 NELSON'S COMMON-SCHOOL ARITHMETIC. 1. To find the difference between 107.06 and .213. OPERATION. 107.060 .213 106.847 2. 617.07 41.7106:=? 6. 341. .213 =? 3. 10.06 .9092=? 7. .97 .0376=? 4. 36.84 6.672 =? 8. 4.15 .1999=? 5. 118.09 7.009 =? 9. 7.96 .9789=? Totals, 351.6056, 725.7592, 352.6506. MULTIPLICATION OF DECIMALS. 217. In this rule we multiply as in whole numbers, and mark off as many places of decimals in the product as there are in the two factors. 1. To multiply 6.107 by 5.5. 6.107 There are three places of decimals in this factor, 5.5 and one place in this; 30535 30535 33.5885 so we point off four in the product, PROOF. 6.107X 5=30.535 6.107X.5= 3.0535 33.5885 EXPLANATION. The 6 and 107 thousandths multiplied by 630 and 535 thousandths, but multiplied by .5 or 5 tenths, it is only one tenth as much, or 3.0535 (Art. ^13), which, added to the first product, gives 33.5885, as above. 2. .3507 X 10.09 =? 7. 2300.7 X 48.003 =? 3. 17.07 X200.6 =? 8. 704.23 X .0007=? 4. 785.4 X 36.70 =? 9. .786x100 =? 5. .279 X160.7 =? 10. 4.862X .75 =? 6. 876.5 X .780=? 11. 200.03 X .002 =? Total, 32980.4659. Total, 110523.641621. DECIMALS 211 218. When the product contains fewer figures than there are decimals in the factors, the number is made up by pre- fixing ciphers. 12. 100X.0005=? 100 .0005 500, to which prefix one cipher and we have .0500, or .05, the answer. PROOF. .0005X100=.05. 13. .107 X-05 ==? 16. .3045X.00061=? 14. 61.04 X-0007=? 17. .27 X-27 =? 15. .7103X-004 ==? 18. .4102X-1004 =? Totals, .0509192 and .114269825. DIVISION OF DECIMALS. 219. When dividing decimals, the quotient and divisor must contain as many places of decimals as the dividend. 1. 33.5885-=-6.107=r? 6.107)33.5885(5.5 PROOF. This is the converse of Ex. 1 in 30 535 multiplication, the multiplier and multipli- cand being 6.107 and 5.5 and the product _ 30535 A further proof is obtained by estimate, if we divide the whole number (83) of the dividend by the whole number (6) of the divisor, which will give one pla-ce for whole number. 220. When the dividend does not contain as many decimals as the divisor, ciphers may be annexed to make up the number. The quotient will then be a whole num- ber, as it simply shows the number of times the latter is contained in the former.* *In practice, decimals are seldom carried to more than four places. 212 NELSON'S COMMON-SCHOOL ARITHMETIC. 2. 3066-r-.1783=? .1783)3066.0000(17195.73752 1783 12830 12481 3490 1783 17070 16047 10230 8915 13150 12481 6690 5349 In this example four ciphers have been an- nexed to the dividend, to correspond with the number of decimals in the divisor. From this resulted 17195, the quotient. These ci- phers being exhausted, five more were an- nexed to the remainders to give the five deci- mals in the quotient. Another method is to annex ciphers at will, observing to place a mark in the divi- dend to limit the whole numbers in the quo- tient, as 3066.0000000. 13410 12481 9290 8915 3750 3566 ~184 221. When there are not figures enough in the quotient to make up the number of decimals in the dividend, ciphers should be prefixed to the former. 3. Divide 10.70067 by 370.4. 370.4)10. 70067(.0288 Here the quotient produced only three 7 408 figures (288), which, with the one in the . divisor, makes only four decimals, so to make U P tn e number a cipher is prefixed. 32947 29632 DECIMALS. 213 Carry out the following to only four places of decimals : 4. 314.06 ~ 10.73 =? 8. 6.742.34 ==? 5. 17600 -f-785.4 =? 9. 496 -*- .278=? 6. 3170.09 -r- 2.4014? 10. 7.6 -5- .734=:? 7. 417.456-5- 31.145 ==? 11. 7.23--4.06 =? Totals, $1385.1824 and 1799.1878. 12. 30.640 -493.67 = ? 16. 724.1 -f-38.07 =? 13. 10.8739-^-117.406=? 17. 82.03 -~- 9.0002 =? 14. 6.342 ~ 22.973=? 18. 7.624 -5- 2.001 =? 15. 1467.06 -=-196.04 = ? 19. .5213-7- .24121=? Totals, 7.9142 and 34.10573. REDUCTION OF DECIMALS. 222. To reduce a Common fraction to a Decimal. 1. Reduce J to a decimal. 2)1.0 By annexing a decimal point and a cipher, the number 7 is properly reduced to tenths or 10 tenths, in which 2 is contained 5 times. This 5, being of the same denomi- nation of the dividend, is tenths, or .5. 2. Reduce J to a decimal. 3)1.00000 This quotient may be carried out indefinitely, oqo oq an d i g called a repeating decimal. To save writing, a point is usually placed over the repeater thus .3. 223. The fractional value of a repeating decimal may be restored by using 9 instead of 10 as the denominator, as f =f 3. Reduce ^- to a decimal. 7)1.000000000000 142857142857+ This is called a circulating deci- mal and is marked thus : J42857=|ff||-=f Its fractional value is restored in the same manner as that of the .3 in the preceding example. 214 NELSON'S COMMON-SCHOOL ARITHMETIC. Express the following decimally:* 4. J 8. A I*. H 16. A 20. 5. i 9. ff 13. ^ 17.. & 21. 6. j 10. ^ 14. ^ 18. 43 22. 7. f 11. J 15. H 19. ^ 23. Answers: .75, .4, .25, .375, .1923+, .6043, .45, .16, .6875, .416, .2187+, .046875, .9, .475, .00224, .46, 183, .6, .02, .5375, .2. 224. To find the value of the decimal part of a com- pound number, as 0.75 or &0.33J. 24. What is the value in shillings, pence and farthings of 0.345? .345 EXPLANATION. Multiplying .345, that is, 20 fbVu by ^' tlie number f shillings in a ' or 6 ' 900 ^ , [A ^ n ^ ^ d muhiplying the 900 shminga by 12, the number of pence in a shilling, we 10.800 Pence have Uffffi or 10.800 pence. Multiplying 4 the .800 by 4, the number of farthings in a 3.200 Farthings P enn y> we have TSa8 or 3 - 200 or 3 ' 2 farth ' or 6s lOd 34- far. in S 8 - The operation might have been abbreviated by dropping the ciphers on the right. Hence, we have for the result 6 shillings 10 pence, 3 farthings and T 2 or ^. Find the value of 25. .625 of a gallon. 28. .1374 of a ton. 26. .1425 of a year. 29. .0037 of a Ib. Troy. 27. .8323 of a . Answers: 21 T %; 2, 1; 274, 12 T ff ; 16, 7f ; 1, 21 T V 225. To reduce denominate values to decimals. *The plus sign will be used when the decimal can be carried out further. DECIMALS. . 215 30. Reduce 6 shillings 10 pence 3^ farthings to the dec- i nal of a pound sterling. 5) 1.0 The first step in this operation is j\ o 9 q <2 / ^ re duce 7= to a decimal, which f T' gives .2. Prefixing 3 farthings, we 12)10.8 o?* lO^j- pence, divide by 4, the number of farthings f>(\\ R a A 9 cJiV77 ina penny, and obtain .8 of a penny. ^jU ) O.5J Cf U^ g b/illl/. r * . ^ to which we prefix 10 and divide by .345 or i'O.-j^o- 12, the number of pence in a shil- ling and obtain. 9. Prefixing 6 shillings, we divide by 20, the num- ber of shillings in a pound, and obtain .345 of a pound, the answer. The reverse of this process is found in example 24. The pupil can prove his calculations by last Art. 31. Reduce 3 quarters to the decimal of a yard. 32. Reduce 6 Ibs. 3 oz. to the decimal of a cwt 33. Reduce 12s. 6fd. to the decimal of a . 34. Reduce 12 Ibs. to the decimal of a tun. 35. Reduce 1 ft. 3 in. to the decimal of a yard. 36. Reduce 16 oz. to the decimal of a ton. 37. At 56 cents a pound, what will 127 Ibs. 6 oz, of tea come to? 16)60 .375 decimal part of a pound. 127.375 56 764 250 6368 75 7133 000 cents, or $7133. REMARK. This is not strictly a practical question, nor the short- est method of computing the above, the object being merely to show the application of decimals. 38. At $5 for a pound sterling, what will be the value of 16 8s. 10d.? 216 NELSON'S COMMON-SCHOOL ARITHMETIC. 39 to 41. What will be the value of the following sums of money at the saine rate? 167 10s. 3Jd., 19 2s. 6d., 10 10s. 10d. Answers: $95.63, $837.57, $52.72, $82.21. XXV. COMMON FRACTIONS. MULTIPLICATION. 226. A fraction is multiplied by a whole number by simply multiplying the numerator without altering the denominator. [Art. 199.] | X 7=7X3, or ^-, which, reduced to a mixed number, equals 5J. 227. Fractions can also be multiplied by dividing the denominator, without altering the numerator. [Art. 202.] AX5=f$=i, or If Multiply the following fractions: 1. |X 5=1J 4. AXH=2-357 2. |X 4=3 5. ^x 9=3.316 3. |X 12=8 6. 7 8 T X 6=2.824 228. Mixed numbers may be multiplied like compound numbers. 7. Let it be required to multiply 4f by 7. Whole Nos. Eighths. Illustration. 4 5 EXPLANATION. Seven times 5 eighths 7 equals 35 eighths, or 4 whole numbers ~~^ ; o anc * 3 eighths. Seven times 4=28 and on* 4 make 32. Ans. 321. or 32f It will not be necessary for the pupil to write his work in so formal a manner as in this illustration. FRACTIONS. 217 8. Multiply 6 by 12. 6-J Multiplying 7 ninths by 12, we have 84 ninths or 9 12 and 3 ninths. Then 12 times 6 are 72 and 9 make 81, .. 3 giving for the answer 81J. or 81] 9. 6J X 8=?* 13. 35 1 X 9=? 17. 914 f Xl20=? 10. 7| X 7=? 14. 60|-X12=-? 18. 63 X 15=? 11. 8| X 6=? 15. 45 f X 8=? 19. 127 T \X 20=? 12. 1,^X12=? 16. 64^X 6=? 20. 110 ^-X 14=? Answers: 54, 50f, 13^, 53 j, 109760, 952.5, 387.29, 2543.636, 1542.3, 316, 730.5, 3S7-&, 365.3. 229. To multiply one fraction by another, the numera- tors may be multiplied together for a new numerator and the denominators for a new denominator. 21. f Xf=3r<V or f Here f is multiplied by 3, giving 6 thirds; but this 3 being sev- enths, 6 thirds is 7 times' too much. Multiplying the denominator [Art. 200] by 7 gives JL or ^. 29 2\s 3\s 4\/ 5 1 20 1 sX^Xs-X-g 3TTF These operations might have been abbreviated by what is called cancellation. In the first example, for instance, -| is to be multiplied by ^; that is, the numerator 2 is to be multiplied by the numerator 3 ; but the 2 is also to be di- vided by 3, for -| signifies that 2 is to be divided by 3; therefore, since the 2 is to be multiplied by 3 and divided by 3, it remains exactly the same, and the 3 of the de- nominator is said to cancel or make void the 3 of the nu- merator. In the following operations the canceled figures will be known by having a line drawn through them. * Many answers in this chapter are given in decimals carried to two, three, or four places, as in practice. Occasionally the plug sign is used to show that the decimals may be continued; at othei times the last figure will be found increased. 218 NELSON'S COMMON-SCHOOL ARITHMETIC. 2 $ 2 In the second operation, the 2 of the nu- I 7~7 merator and the 6 of the denominator are uncanceled, making f, which, reduced by _X-X-X- dividing both by 2, equals J. $^06 The 2 and 6 might have been canceled t $ $ a ^ s 5 ^ dividing both where they stood in -X-X-X- the question, as in the third example, plac- ing only 3 as a denominator, and 1 as a nu- merator. 1 is always to be understood where a number has been canceled. 8 23. i 1 Some prefer arranging the terms of can- 1$ 3 celing fractions as in the margin, with 15 the denominator or divisor on the left, "^ and the numerator on the right. EXPLANATION. The first 2 was canceled in the 18, leaving 9; the 24 and 9 were canceled by dividing both by 3, leaving 8 and 3; the 74 was canceled by the second 2. 24. iX -I X f =? 28. 1JX 25. fX * XifX T V= ? 29. 9K Is/ 8 N/ 5 \/ 7 ? 30 ^O. -^X T5"^34<^2T 27. |XHXfXj=? 31. Answers: 3.375, 4.8, |, J, .05, 5.75, .0041, .0264+. 230. Compound fractions are reduced to simple ones by multiplication. Let it be required to reduce \ of of | to a simple fraction. We know, by inspection, that one- half off is ^, and that \ of f is J 3 .the answer. By multiplication: By cancellation: -XTX-= ?> I, Jl FRACTIONS. 219 32. of | of ^=? 36. | of f of 9J =? 33. * of # of 3*=? 37. fof }'<&& = ? 34. f of % of -&=? 38. 4 of li of =? 35. of Ij-of =? 39. iof |x*Xli=? Answers: 7.389, 1.75, .0068, .286, 1.5, -J, .4286, .3. 231. To multiply one mixed number by another. The mixed numbers are reduced to improper fractions and multiplied as in Art. 229. 40. 3J by 5*=? 41.44X71=? 44. 4 T * r x3|=? 47. Ifx 8=? 42. 2JX1=? 45. 2JX f=? 48. 72|x62>=? 43. 3|X2 T 5 T =? 46. 6g- X2^=? 49. 87-5X16?=? Answers : 29f, 17^, l^J, 8||, 2.7, 1^, 17*i 4537.5, 1458.3, 4.8, 14.583. 50. At 11| cents a pound, what will 147J Ibs. of coffee cost? 51. What will 7^ Ibs. of cheese cost, at 9J cents per pound? 52. At 12J cents a pound, what will 120 Ibs. of sugar cost? 53. What will 14^ Ibs. of beef cost, at 6j cents a pound? 54. Fifteen and a half yards of muslin, at 9^ cents, will cost how much? Answers: 71 ts, $14.80, $16.59, 98 c, $1.43, $16.50. DIVISION. 232. Division being the reverse of multiplication, to divide a fraction by a whole number, we divide the nu- merator or multiply the denominator. [Art. 2001.] 220 NELSON'S COMMON-SCHOOL ARITHMETIC. 2.. -j-g 4=? 5. --- 7=? ;. ^^-9=?- . A-s-6=? 6. ^.e^io^? 9. 4. J8. 3? 7. I8_i_ 4? 10 " 7 - 9=? 36 * ^Y * * ~~ 1 * Answers: ^, gL, ^ .1887, .076, .16, .16, .125, f. 233. To divide mixed numbers. 11. 21|-H-6==? Whole NOB. Fifths. EXPLANATION. 6 is contained in 21, 3 times '_ __ an d 3 ^ e ft" l n tne & f tne remainder there are 3 3 15 fifths, which, added to the 3 fifths in the ques- or 3-| tion, make 18 fifths. % 6 in 18, 3 times. Ans. 3. 12 1 2 ' 8 ? 8)124- EXPLANATION. In this example we had 4 remainder, 129 in which were 28 sevenths, and the one in the question & made 29. Then, as 8 would not divide 29 without a re- mainder, we multiplied it into the denominator, which made 56. Ans. 1||. 13. 67f-r- 7=? 17. 167^25 = ? 21. 72f-- 9=? 14. 445 _,_ 3== ? is. 21J--14=? 22. 148i-*-27=? 16. 118^12=? 20. 22J-i-12=? 24. 175f-v-15=? Answers: llf 8^-, 5.5, 9f, 14||, 6.684, 1.523, 2.306, 23.05, 11.7142+, 19||, 9, 1.861. 234. To divide one fraction by another. OR i . 3 ? ft9 2~*~? 1 . 3 1 V 4 4 ^ 2 j-s-f =2 X -3 ^ B^- Multiplying the denominator of the dividend by 3, we have \ [Art. 200-1], but as the divisor is not 3 but 3 fourths, we multiply the result by 4, giving |-, or 1^-; hence, we divide one fraction by another by inverting the terms of the divisor, and multiplying as in multiplication.* 235. To divide by a mixed number. * CAUTION. The pupil will observe not to invert the terms of the number to be divided. FRACTIONS. 221 26. 369--97rr=? Hero both terms are reduced to the same denomi- *4 obJ nation (thirds), and division performed as in whole ~ numbers. 97J is contained in 369, 3|| times. ^292 )H07(3f|i 876 231 27. 76-f-2| = ? 30. 349 ~ 4J=? 33. 73^8 f =? 128. 84-^-3|=? 31. 106f-7- 5|==? 34. 73 -=- ^=? ^29. 82-r-7f=? 32. 276f-f-12 = ? 35. 191 1 -- T \=? Ans. 28fft, 23.05, 29.37, 10*f, 77.5, 18 J^, 8.565+, 719^, 293/254. 236. Complex fractions are unsolved questions in di- vision. 36. 2J 7 37. f-f-f =? 41. i of f -f-f =? 38. f-^|=? 42. l|xiH-| =? 39. ^-f- 1 = ? 43. 2J X T-4-| of |== ? 40. i-H-i=? 44. 3^ * \/ . O rv-f- 7 f s\ ~^~ ' >? ^-*- "S"~~~ 45. If 120-J Ibs. of cheese cost $14.80, what will 1 Ib. cost? 46. Find the cost of 1 Ib. of coffee, when 15^ Ibs. cost $1.43. 47. If 11^ yards of cassimere cost $16.59, what will one yard cost? 48. If 9-J yards of muslin cost 71 cents, what will 1 yard cost? Answers: 7^, 9^., 147/ 3 , 12^ r , 1.185, .6, .3, .53, .36, ., 5$, 2.571. 222 NELSON'S COMMON-SCHOOL ARITHMETIC. SUBTRACTION. 237. To subtract fractions or mixed numbers from wlwle numbers. 1. From 87 take 25f Whole Nos. Sevenths. TO subtract 3 sevenths, one whole number " is added to both terms. In 1 there are 7 sevenths, from which subtract 3 and the re- 61 4 mainder is 4. 1 to the 25 makes 26, which take from 87 and 61 is left, giving for the answer 61 A. The above formula being used for illustration only, the learner will be expected to write his operation as follows: 87 25* ~61|~ 2. 210371=? b. 1003819 * ==? 10. 2925 J =? 3. 119 82-J? 7. 3785 10 -f- = ? 11. 167 89f = ? 4. 61 4f = ? 8. 2168 14 f = ? 12. 3621^=? 5. 54_ 51=? 9. 1765 777|=? 13. 218 36 1=? Answers: 987.4, 2153.571, 3774.14, 172.5, 36.875, 56. 1, 48.875, 243.8, 172J, 3.25, 77.125, 181.625, 14|f, 3.75. 238. To subtract a fraction from another of the same denomination. 14. From take *. 64=2 or I 6 sevenths less 4 sevenths leaves 2 sevenths, the answer. 15. |-i=? 18. |i-|| = ? 21. 16.1-|=? 19. |_4 =? 22. 17. i 3 2 - T V=? 20. ^- 3 \=? 23. Answers: J, II, .2, .25, .4, .14285+, ft, -i, .1, f 239. ^b subtract one fraction from another, both terms must be of the same denomination, or reduced to the same FRACTIONS. 223 denomination. | can be subtracted from f , but J can not be conveniently subtracted from | without first changing the denomination of one or both. 24, Subtract J from J. EXPLANATION. The 6, being common to both fractions, is called the common denominator. To raise two fractions to a common de- nominator, both terms of each fraction may be multiplied by the denominator of the other. [Art. 197.] oc 3 2 9 ^?ft 4 1 ? Q"l 3 ___ 1 ^ rt/> 4^ i o on 2 5 y oo fj 1 9 97 7 1 O Oft 7 4 > oo 2 1 ? Ail I TT 15" 4 OV. "If ?" * OO. 7j y -|-7j I Answers: |f, .2576, .3035+, .3, .07, ^, .375, .0396+, .075, .3096. ADDITION. 240. Fractions of the same denomination are added to- gether by finding the sum of the numerators. 1- l+f+l-2+5+7 thirds, or 4J. s' A+A+A+i-? Answers: 1.336, .851, {^J, J, 1 7 \, 2 T 4 T , 4 T \, 4|. 241. Fractions of different denominations are added together by finding a common denominator, as in subtrac- tion, and proceeding as in last Art. 8. Find the sum of f+f+f. 3X5x6:=90. Common denominator. 1 \ 2X5X6= 60 Here the numerators 2, 3 and 5 are multi- 'I 3X3X6= 54 plied successively by all the denominators but | 5X^X3= 75 their own. 2, for instance, is multiplied by -j oq &X6} the product of which is also the multi- I fpj . 189 Q ^ plier of the denominator 3, giving for the . 1S "^" -^TTT- fi rs t fraction |g instead of J. In the same way | becomes |J, and | Jg, making 189 ninetieths, or 2.1. 224 NELSON'S COMMON-SCHOOL ARITHMETIC. When there are many fractions to be brought to the same denomination, it will be better to first divide the common denominator by each denominator. This gives the number which will raise both terms to the required denomination.* [Art. 197.] 9- B- 4+ i=2 13. 8 +6 +12 =? 10. H- 4+ i=? 14. 2$+ JH- if= ? 11- !+ t+ 7T= ? I 5 ' Fr+ 9 + TT= ? 12. 6J+7f+8=? 16. 6f +lf + 2 = ? Answers : 7J, 10.3714+, 1^, .4345+, 1.0707, 9.612+, 32.25, 3.995+, 22.277, 26.75. 19- 20. 21. | Of f +| Of 22. |X T 9 ir+ of fX? of +l+5f +J=? 23. 2|X6i+8+f of t+l of 3+7=? 24. lX2+ of of *THE LEAST COMMON MULTIPLE Of several denominators is the least number which can be divided by them without a remainder. The following is the process for finding it: To find the least common multiple of 3, 4, 6, 8, 9, 12, 15. 2)3 4 6 8 9 12 15 EXPLANATION. 2 was used as a di- 3^3 2 3 4 9 6 HT visor of 4, 6, 8 and 12, and the. quo- 2 77 ~ - ~ ~ ~ ~ tients set down. The other numbers were brought down and 3 divided into those divisible by it without a rernain- 36 - der; and so the process was continued until no number could be found to divide the others without a re- mainder. The divisors and remaining numbers then being multi- plied together, produce 360 as the least common multiple. Like the operose method of finding the greatest common divisor, this is seldom used. FRACTIONS. 225 25. 4f x5J+6iX2| of 26. | of f +? of I of 2T. 28. 29. f of f + _^ + 1 of ^ =1 30. 4- + ^ of }f + | of 5^=? 31. |f of7f of 9 + ~ of 14 ==? 32. + |iof2f + |- of 6f=? 33. ^of i|of9|+^of 8=? 34. + + ii+e =^ Answers: 14^, 15/ T 8 3 +, 34.91+, 85.9146, .01269, 7.14285, 26.1428, 5.25, 141.85, 175.85, 1.9642, 12 J, ^W^ 3.4531+, 14.183+, 4.350+, 53.83, 11.168+, 3.0278+, 4.0126+, 32.2678+, 11.694+, , 82.61. PRACTICAL QUESTIONS. 1. In an invoice of goods there are the following items; required the amount. 27 J doz. @ 9Jc 13f doz. @ 5^c. 18 T 7 ^ " " 12 16| " 3J 16^ " 12J 118^- 2J -4ws. $10.65. 2. of a merchant's goods were destroye^ by fire, and what remained was worth $1637.50; what was his loss? 3. A owns of a steamboat, B J and C the remainder, which is worth $10000; what is the value of the boat? 4. J of a saw-mill belongs to A, to B, T 3 ^ to C, the remainder to D, and the profits for the year amounted to ; $1680; what is each man's share? 5. The par value of the pound sterling is $- 4 -; required the value of 1674 at 10% premium. 226 NELSON'S COMMON-SCHOOL ARITHMETIC. 6. A can do a piece of work in 8 days, B in 7 days and C in 6 days ; in what time can they do it if all work together? SOLUTION. A can do , B -f and G J of the work in a day. The sum of these fractions is -$%. If y^ can be done in a day, -J-JJ! (the whole) can be done in J^=2^-|-, or 2 days 3^ hours. 7. There are 3 pumps placed in a coffer dam ; one will empty it in 10, another in 15 and a third in 20 hours; in what time can it be emptied by working all three at once? 8. Express ^ of a day in hours, minutes, etc. SOLUTION. | of a day is the same as -| of 3 days. Days. Hours. Min. Sec. 7)3 00 10 17 8$ EXPLANATION. As 7 is not contained in 3 days, we reduce them to hours=72 hours, which, divided by 7=10 hours and 2 left, etc. 9. In of a pound (British money), how many shil- lings and pence? 10. In |- of a bushel, how many pecks, quarts, etc.? 11. In \ of a ton (long weight), how many hundreds, etc.? 12. Find | of 167 18s. 6d. 13. Eeduce ^ an inch to the fraction of a foot. 14. Reduce -^ a cent to the fraction of a dollar. 15. What part of a pound Troy is \ an ounce? 16. What part of a ton is J of a pound? 17. | of a farthing is what part of a pound? 18. | of f of $1600 is what part of $1000? Answers: $420; $210; $315; $735; $480.00: $3275; 16s. Bd.; 3 pks. 4 qts.; 3 cwt. 1 qr. 9 Ibs. 5 oz.; 4 hours; $8184; y^; ^ ; 3 ^ ; f; 3^; -g-^. PROPORTION. 227 RATIO. 242. The relation that one number bears to another is called ratio. The quotient arising from dividing one num- ber by another, of the same denomination, is the ratio between them. And as two quotients can be obtained from comparing any two numbers, it follow* that two ratios can also be obtained. The relation that 1 bears to 2 is J, and that which 2 bears to 1 is f . The sign of ratio is the colon. The above ratios would be expressed thus: 1 : 2 and 2 : 1, and would be read one is to two and two is to one. Some mathematicians divide the first term by the second; others, the second by the first. The former method is most used: 6 : 3 will equal | or 2, f : ^=i=j, or 1|. i 243. Numbers or quantities of different denominations can not have a ratio. We can not compare 3 trees with 5 books. But if the numbers are capable of being re- duced to the same denomination, they can be compared ; for we can say 3 feet is to 2 inches, as it is the same as to say, 36 inches is to 2 inches. XXVI. PROPORTION. 244. Two ratios may be equal to each other. 2 : 4~ :8. 2 bears the same relation to 4 that 4 does to 8. 245. When ratios are equal, the numbers or terms which compose them are said to be in proportion, and are written thus : 2 : 4 : : 3 : 6, and read 2 is to 4 as 3 is to 6. 228 NELSON'S COMMON-SCHOOL ARITHMETIC. The first and last terms, as the 2 and 6, are called ex- tremes, and the second and third the means. 246. The same ratio may arise by comparing 4 quan- tities, two of which are different in denomination from the other two. Tons. Tons. $ $ 3 : 6 : : 6 : 12. The ratio is J, and if reversed, as 6 : 3 : : 12 : 6, it would be 2. 247. If the extremes are multiplied together the pro- duct will be equal to the product of the means. 3X12=36 6x 6=36 Hence, when any 3 terms are given, we can readily find the fourth by dividing the product by the odd term. If we had only the three first terms of the above proportion, Tons. Tons. $ that is, 3 : 6 : : 6, the fourth term would be found by dividing the product of 6X6, or $36, by 3=$12, or thei fourth term as above. To apply this in practice, we have only to suppose the 3 tons and 6 tons to be coal, and the $6 the price of 3 tons. Then 3 tons is to 6 tons, as the price of 3 tons is to the price of 6 tons. 1. What will 35 Ibs. of sugar cost, if 7 Ibs. cost 77 c.? STATEMENT. 7 : 35 : : 77 is to the price of 35. 7 : 35 : : 77 EXPLANATION. The means, 35 and 77, 35 ing multiplied together, produce 2G95, and ~^ r this divided by the given extreme, 7, gives ~~.j the required extreme, 385, which must be of ., the denomination of cents, in order that a 7)2695 ratio exist between it and the third term ~385 cents, ceuts - or $3.85. PROPORTION. 229 The same by cancellation. 5 =385, or $3.85, Am. By placing our terms in fractional form, we have 35X77 for a numerator and 7 for a denominator. Then reducing both terms in the same ratio, the 7 cancels the 35, leaving 5X77 for a numerator and 1 for a denominator. 2. If 27 Ibs. of butter cost $3.75, what will 16| Ibs. cost? 3. Find the price of 12^- dozen of chickens, at 30 cents a pair. 4. The price of 21 tons, 13 cwt., 3 qrs. and 15 Ibs. of hemp is $1680.55; what will 15 cwt. cost? 5. What will 54 Ibs. 7-J- oz. cost, if 15 J Ibs. cost $8.47? 6. If f of a ship cost $7000, what will T 9 a cost? 7. If 6 men do a piece of work in 7 days, how long will it take 5 men to do it? STATEMENT. 5 : 6 : : 7 The 7 (days) having no ratio to the 7 other numbers of the proportion, is 5VL9~ placed to the right. At first sight, it ' would seem that the proportion would 8 1 be 6 : 5 : : 7; but 6 men do not bear the same ratio to 5 men that the time of the 6 bears to that of the 6. A little reflection will convince the learner that 5 men would I require a longer time to do the work than 6 men, which would fail j to complete a proportion, as shown by the following j STATEMENT. The greater : the less : : the less : the greater, or, the greater X the greater=rthe less X the less! Hence, to find the second term of a proportion, it will jbe necessary to inquire whether more or less will be re- | quired. If more, put the greater of the two terms in the 'second place; if less, put the less of the two terms in the second place. 230 NELSON'S COMMON-SCHOOL ARITHMETIC 8. If two men plow a field in 3 days, how long will it take 3 men to do it? 9. If 26 yards of linen cost $13.50, what will 10 yards cost? 10. If 3 coats can be made from 10J yards of cloth, how many can be made from 31^ yards? 11. If the interest for $750 for 3 years, 4 months and 10 days be $151.25 (360 days to the year), what is it for one year? 12. The interest of 100, from 3d of April to 25th of February, is 6 5s. 9|^(Z. ; what is it per year? 13. A, B and C are in partnership, and their gains for the year are $6757 ; what is each man's share, suppose A invested $1567, B 2600 and C 3798? The sum of their investments is to each man's invest- ment, as the total gains to each man's gain. 14. M invests $6500, N $1487, $3654; in 3 months it is found that their gains are $1678; what is each man's share ? 15. A lends B $1000 for 13 months 10 days; how long should B lend A $8271, to return the favor? 16. If the shadow from a two-foot rule be 6 inches, what is the height of a tree that throws a shadow of 75 feet? 17. If 7 men can build 21 perches of masonry in a day, how long will it take 14 men to build 147 perches? Answers: $5.19, 9, 3, 7, 300, $1329.34, $936.89, 49, $5.90, $45, $4.50, $2.25^, $22.50, $7350, $58.09, $30.25, 3,2. REMARK. This rule is of less utility to the business man or me- chanic than is generally claimed for it, as most of the problems can be solved in less time, and with fewer figures, by the application, of Multiplication and Division. Take, for instance, the 17th. If PROPORTION. 231 21 perches can be built in a day, 147 can be built in 1477 days; and if 7 men can do it in 7 days, 14 men can do it in one-half of 7, or 3J days COMPOUND PROPORTION. 248. A proportion is said to be compound when it is composed of more than two ratios or four terms. 1. If 3 men in 5 days, by working 8 hours a day, dig a cellar 15 feet long, 12 feet wide and 7 feet deep, in how many days will 2 men dig one 17 feet long, 14 feet wide and 6 feet deep, by working 10 hours a day. In this problem there are 11 terms and 5 ratios: the ratio between men and men, that between hours and hours; between feet and feet of the length ; feet and feet of the width, and feet and feet of the depth. In arranging these terms, we proceed as in simple proportion, Ex. 7, page 229. Days. Men, 2 : 3 : : 5 1. Days are wanted; write days as Hours, 10:8 right-hand term. Length in ft., 15 : 17 2. Comparing men with men, we Width in ft, 12 : 14 find that it will take 2 men a longer Deptli in ft., 7 : 6 time to do the job than it took 3 men, BO we write the greater of the two terms (3) in the second place. 3. Comparing hours with hours, we reason that it will take less time to do the job by working 10 than by working 8 hours a day, BO we write the smaller number on the right and under the second term. 4. Comparing length with length, we reason that it will take a longer time to dig a cellar 17 feet long than it did to dig one 15 feefc long, so we write the greater (17) term under the second term. 5. Comparing breadth with breadth, it will take a longer time to dig a cellar 14 feet wide than it did to dig one 12 feet wide, so we write the greater (14) under the second term. 6. Comparing depth with depth, it will take less time to dig a cellar 6 feet deep than it did to dig one 7 feet deep, so we write tho smaller numbei under the second term. 232 NELSON'S COMMON-SCHOOL ARITHMETIC. The same by cancellation. 17x2 =6$ or 6 days 8 hours.* $ $ 17 REMARKS. 1. The pupil should observe to have the terms of each ratio of the same denomination. 2. The answer will be of the same denomination as the right- hand term. 2. If 6 men in 15 days dig a trench 18 feet long, 7 feet wide and 5 feet deep, in how many days will 21 meu dig a trench 125 feet long, 9 feet wide and 4 feet deep? 3. What is the interest of $6784 for 2 years, 6 months and 15 days, at 6% per annum? STATEMENT. Days, 365 Dollars, 100 $6 927 6784 Ans. $1033.77. 4. The interest of $1467 for 3 years, 4 months and 12 days is $450.72; what is the rate per cent.?f 5. The interest on $786.55 at 10% is $176.44; what is the time? 6. The interest for a certain sum of money for 4 years, 2 months and 20 days, at 6%, is $100; required the prin- cipal? *In forming this proportion, we reasoned from what was given to what was required. For instance, in comparing men with men, we inquired if it would take 2 men a longer or a shorter time than the time (5 days) that it took 3 men to do it. fThe pupil can prove his own work by computing the interest by the method taught in the first part of this book. PARTNERSHIP* 233 XXVII. PARTNERSHIP. 249. WHEN two or more persons associate together to carry on a business, they are said to be in partnership, and are called a firm, house or company. Partnerships may be general or special. General part- nerships extend to the whole of the mutual dealings of the parties. Special partnerships are formed for some specific purpose, a single dealing or adventure. When more than two persons are engaged in business, it is usual to select the names of one or two of the mem- bers, with the term U 0o.," for the name of the partner- ship; as a business conducted by Messrs. Jones, Evans, Henderson and Norton might be called the firm or house of Jones & Co.* 250. Each member of a firm becomes responsible for the acts and contracts of his copartners, in the way of sale, purchase, promise, agreement, etc., performed in the course of the usual business of tfce firm. If a partner draws a note or bill even in his own name, and it can be proven to be on account of the partnership business, he 1 thereby renders the firm liable. 251. An individual becomes a partner by allowing the community to presume that he is such, or by having his name appear on a sign or in a bill, card, etc. A secret partner becomes equally liable when discovered, as if his name appeared in the firm. 252. A creditor of one of the members of a firm can *The name of a firm is not always derived from the members having the largest interest. Where precedence is not given to age, the names of the most influential are usually selected. 20 234 NELSON'S COMMON-SCHOOL ARITHMETIC. claim only the interest of the debtor in the partnership property after all claims against it have been settled. 253. All partnership agreements should be written. The funds, property and merchandise furnished by partners for carrying on business are called stock or cap- ital, and the gains are called dividends. The liabilities of a partnership or individual business are the debts, and the assess, their available means, includ- ing the indebtedness of others to them. An inventory is a list or statement of those things which constitute assets. 254. In keeping partnership accounts, each member of the firm should be credited with all that he brings into the concern or business, and be charged or debited with all that he takes out, just the same as if he had no inter- est in it. 255. The calculations peculiar to partnership relate to the division of property and profits. 1. A, B and C have been in business one year, and find they have made a net gain of $3476, which is to be divided as follows: A is- to have J, B J and C J; required the share of each. $Mp._$i738 j A's share; 8^-&=J869, B's share; and $869=C's share. 2. X, Y and Z purchase a tract of land for $2000, X giving $600, Y $900 and Z the remainder. In one year afterward they sell it for $5500; required each person's share of the proceeds. 3. A, B and C invest $2000 each. In three months their gross gains are $2000 ; expenses^ including $250 for additional services of C, $600: what will be each man's share of the gain? V . JOINT STOCK COMPANIES. 235 XXVIII. JOINT STOCK COMPANIES. STOCKS. 296. A Joint Stock Company is a body of men asso- ciated together in a species of partnership, to carry out some heavy undertaking requiring the investment of more capital than individuals or partnership companies com- monly possess. Joint stock companies are usually ineor-^ porated by act of legislature, with certain privileges. Railroads, canals, bridges, etc., are generally constructed by this species of combined interest, and many banking and insurance houses, scholastic institutions, etc., are owned and managed by joint stock companies. When an association of this kind is to be formed, a few leading persons make an estimate of the probable amount of capital required, divide it into equal shares of from $10 to $100 or $500, according to the nature of the un- dertaking, and issue certificates of ownership for each share. These are called certificates of stock, and are trans- ferable. Persons owning certificates are called stock- holders. Joint stock companies are usually managed by a presi- dent and board of directors^ elected for the purpose, by the stockholders. When shares sell for the price named in the certificate, the stock is said to be at par; if above this value, they are said to be above par; if below it, below par. Besides the stocks of companies, there are government stocks, which consist of bonds that have been issued by state officers, for the purpose of borrowing money. These draw interest at a specified rate. 236 NELSON'S COMMON-SCHOOL ARITHMETIC. In dividing the profits of joint stock companies, it has been found more convenient to declare the dividend by percentage. 1. What is the cost of 10 shares of railroad stock at 5% below par, the original cost being $100 per share? Find the cost of 10 shares at $100 and deduct 5. 2. A banking institution declares a dividend of 18% on a capital of $30000 ; what amount of money should a stockholder receive who holds 5 shares, valued at $200 each? 3. I hold 15 shares (each $100) of stock in gas-works, which have declared a dividend of 20%; how much am I entitled to after my gas bill of $20 is deducted? 4. How many shares of United States stocks at 2% above par can I buy for $1224, the original cost being $100 per share? 5. What amount of stock can I buy for $1683, if I am allowed 2% commission on the amount invested? The amount I am to receive is to be y-g-g- or ^ of the amount of stock purchased not -^ of $1683, for that would be commission on commission and investment. Let the amount to be invested be represented by |-g, and to this add ^j==|^, then we discover that $1683 is f-J- of the amount to be invested. 1 18 3 33=-^, or my commission, which, if we multiply by 50, will give us the amount to be spent, $1650. To prove this, find the commission on $1650 at 2%. 6. A broker receives 1685, which he is desired to in- invest in State stocks; how much should he invest, and allow himself 2J% on the investment? 7. What amount of stock can a broker buy for $16700, and allow himself J% on the investment? Answers: $180, $950, $280, $16658,35, $6619,51, 12. JOINT STOCK COMPANIES. 237 257. When declaring dividends, it is customary to re- serve a part of the gains of business for current expenses. Such sum is carried to an account called a Contingent Fund or Contingent Expense Account. Dividends, in this way, are usually declared for an even rate per cent. 8. A coal oil company, with a capital of $150000, gain! $31,493, and has concluded to declare a dividend of 15%: how much will be left to apply to contingent fund? 15% of $150000=$22500. $31493 $22500=$8993, Am. 9. An insurance company gains $53,369.87, in six mos., on a capital of $500000, but does not consider it safe to declare a full dividend; how much will apply to contin- gent fund account after declaring a rate of 10% per an- num? 10. What is the largest even dividend which can be declared by a company with a capital of $150000, whose gains are $13547.65? 11. A stockholder, owning 20 shares, at $50 each, re- ceives a dividend of $120; what is the rate per cent.? 12. The first dividend of a company is payable in bonds, by which a stockholder, owning 20 shares, obtains two shares worth $100 each; what was the rate of dividend declared? 13. What would apply to contingent account where a dividend of 20% was declared on a gain of $316784.87, arid a capital of $2000000? Answers: $116784.87, 10%, 12^, 9%, $28369.87. 238 NELSON'S COMMON-SCHOOL ARITHMETIC. XXIX. BANKRUPTCY INSOLVENCY. 258. Bankruptcy signifies inability to pay. A person becomes bankrupt when he is obliged to give up his busi- ness for want of means to pay his debts, and to carry it on. Such an individual is said to have failed. Bank- ruptcy and insolvency are synonymous terms. Insolvent debtors usually transfer their property to other parties for the benefit of their creditors. This is called making an assignment, and prevents the individual debtors from recovering more than a share of the property apportioned to the amount of their claims. The person to whom an assignment is made is called an assignee, the property and claims of the debtor, his effects or assets, and his indebtedness, his liabilities. I. A person failing in business has the following effects to meet claims to the amount of $13000; how much should his creditors receive on the dollar? Merchandise to the amount of $3500, railroad stock to the amount of $2100 and personal claims to the amount of $1500. 3500 2100 1500 Amount of assets, 7100, which, reduced to cents, and di- vided by the amount of the liabilities 5 4 T 8 ^ cents, or *13|000)710|000(54 T 8 3 65 52_ ~8~ IMPORTING. 239 2. The amount of assets belonging to an insolvent debtor is $4684, and his liabilities $22000; how much can he pay on the dollar? XXX. IMPORTING. 259. IMPORTING is the business of buying goods in a foreign to sell in a home market. A tax, under the name of duties or customs, is imposed by government on most imported articles of commerce. Such taxes are levied for the purpose of creating revenue to defray the expenses of government or to protect home manufactures and agricul- tural interests. Duties are regulated by a scale of prices called a tariff] and are altered according to the exigencies of the times or caprice of the administration. A liigli tariff signifies high rates of duties, and a low tariff, low rates of duties.* The persons appointed to examine imported goods and collect taxes are called custom-house officers, and their place of business, the custom-house. 260. Duties are of two kinds: Ad valorem and Specific. Ad valorem duties consist of a rate per cent, on the value of goods as stated in the invoice; Specific duties, of a stated sum of money on the quantity imported, without re- gard to value, as $1 a gallon, $20 a ton. 261. Certain allowances, called draft, tare, leakage and breakage, are made on goods charged with specific duties. These allowances sometimes consist of a percentage of the weight or quantity and sometimes of a specific deduction. Tare is an allowance made for the weight of the box, * Taxes are often levied upon exports as well as imports. 240 NELSON'S COMMON-SCHOOL ARITHMETIC. barrel, bag, crate, etc., which contains the goods, and is usually calculated by percentage ; etc., after the deduction for draft is made. Draft or tret is an allowance made for loss by weighing in small quantities, and for impurities to which some goods are subject. On 112 Ibs., or less, it is 1 Ib. From 112 " to 224 Ibs., 2 Ibs. 224 " " 336 " 3 " 336 " " 1120 " 4 " 1120 " " 2016 " 7 " More than 2016 " 9 " Leakage is an allowance of 2% on liquids in casks, paying duties by the gallon. Breakage is an allowance on bottled liquors, usually 5%, but on ale, beer and porter, 10%. Gross Weight is the total weight of goods and box, bar- rel, etc. Net Weight is what remains after all deductions are made. We shall not trouble the learner to work out any ques- tions in this chapter, as it rarely happens that young peo- ple have them to do in business. XXXI. FOREIGN EXCHANGE. 262. IN calculating Foreign Exchange, the money of one country has to be represented in that of another. A bill drawn in New York on a merchant in England will be expressed in pounds, shillings and pence. The draft, though not stated in the question, is to be deducted bo- fore other allowances are made. FOREIGN EXCHANGE. 241 263. The relative value of moneys of different coun- tries depends on the par of exchange and the course of exchange. 264. The par of exchange is the comparative value of the coins of the different countries, and is fixed, while the relative purity of the coins is the same. The par of ex- change between the United States and Great Britain is $4.86 to the pound sterling. Formerly, when the silver of the United States was purer, the par value of the pound sterling was 4|, and exchange is still quoted from this par, or 9^ premium on the old currency being the present par value. 265. The course of exchange usually depends upon the relative state of indebtedness of the merchants of the dif- ferent countries, and the supply of gold and silver; ac- cordingly, the course of exchange will sometime be above and sometimes below par. EXCHANGE WITH GREAT BRITAIN. FORM OP A FOREIGN BILL. Exchange for 1567. CINCINNATI, June 3, 1867. Thirty days after sight of this first of Exchange, (sec- ond and third of the same tenor and date unpaid,) pay to the order of William Tuechter, the sum of One thousand five hundred and sixty-seven pounds sterling, value re- ceived, and place to my account as advised.* To William Morgan, JSsq., J. B. TREVOR. Liverpool, England. * Foreign bills are generally drawn in sets of two, three or four; that is, copies of the same bill are made out and transmitted by different conveyances to the payer, one of which being received and accepted, or paid, the others to be void. These copies are called First, Second or Third of Exchange. The above is a copy of the first. The others are drawn in a similar manner. 242 NELSON'S COMMON-SCHOOL ARITHMETIC. BRITISH OR STERLING MONEY REDUCED TO FEDERAL MONEY. GIVING the cost of British money is called quoting it, and the rate, the quotation. Sometimes the total cost of a pound is given; at other times, only the premium on the old par. 266. When the Amount is given in the Quotation. 1. What is the value of 157 9 2 in Federal money @ $4.86 to the pound sterling? SOLUTION. Since 1 is equal to $4.86, 157 will be equal to 157 times 486, or $763.02. Then taking aliquot parts of a pound, we have 6 shillings and 8 pence:=J, and 2 shillings and 6 pence^^J of a ; add J and -J of $4.86 to $763.02=765.25. TABLE OF ALIQUOT PARTS Of a Pound. G>/ a Pound. 0/ a Shilling s. d. d. d. 10 is i 10 is 7 V 6 is 6 8 8 i sir 4 4 5 I 7J 3 i 4 6 5 2 3 4 5 A I* ^ 2 6 i 4 1 T 2 3 A Of 1 8 _C 2 0^ ^4 1 4 IT 1J yi(5 < 1 3 A 1 T0 1 7>Tf 2 to 4. At $4.84, what will the following sums amount to? 345 14 0, 15 3 4, 365 7 6. 5 to 7. 147 13 4 @ $4.87, 425 17 6 @ $4.44, 652 10 @ $5=? Answers: $5872.53, $3515.02 267. 2b reduce British to Federal money when the Pre* mium only is given. FOREIGN EXCHANGE. 243 8. What is the value of 221 15 6 @ 8% premium? The premium is reckoned on the old par value; that is, $4.44| to the pound, or $40 to 9. 12)6.0 Reducing 15 shillings and 6 pence to the decimal of a P oun< *j we have -775. 775 Hence, 221 15 6=r-221.775, which, multiplied by 40 and divided by 9=$985.666, to which, if we add 8%, we shall have for the answer, $1064.52. What is the value of the following in Federal money? 9. 1424 19 9 @ 7J% 13. 313 8 4 @ S%% 10. 3575 18 6 @ 8 % 14. 505 19 6 @ 9 % 11. 1100 12 6 @ 8J% 15. 3737 12 3 12. 111 @ 9f # 16. 649 4 6 Total, $29793.55. Total, $25270.28. 267. When gold is at a premium, the rate may be added after the rate of exchange. Take the 8th example, assuming gold to be at a pre- mium of 45 cents, or 45%. Value of 221 15 6 at the old par^$985.666. To this add 8% premium on British money and 45% on gold to the sum, and we have $1543.56. 268. TO REDUCE FEDERAL TO BRITISH OR STERLING MONEY. CASE I. When the Amount is given. 17. At $4.87 to the pound sterling, what will be the value of $37654? Since $4.87=1, $37654 will be the equal to as many pounds as $4.87 is contained times in that number. W65400-r-487==7731.827 f which reduced=7731 16 6J. 244 KELSON'S COMMON-SCHOOL ARITHMETIC. Reduce the following to British money: 18. $3674.87 @ $4.87. 19. $67845.18 @ 4.44|. Total, 16019 15 2. CASE II. WJien the Premium only is given. 20. At 9% premium on sterling money, what will be the value of $3964? At 9% premium $1.09 is worth only $1; therefore, $3964 are worth as many dollars as $1.09 is contained times in it. 396400^-109=3636.697, which, reduced to pounds by multiplying by 9 and dividing by 40=818.2568, or 818 5 1J. 21 and 22. Reduce the following to British money : $3165 @ \%, $1678.90 @ 9%. Total, 996 2 2J, XXXII. DTSITRAETCE. 269. INSURANCE is a guarantee against loss. It may be of several kinds, as Fire, Marine and Life insurance. 270. Insurance on fixed property is called fire insur- ance; that on movable property, as goods in course of transportation, ships, etc., is called marine insurance; that which guarantees the payment of a sum of money to a survivor at the death of an individual, life insurance. 271. The act of insuring is termed taking a risk; the amount paid for insuring, the premium; arid the paper upon which the contract is written, the policy. 272. When the risk is heavy, the insurer sometimes re-insures in another company. 273. In time of war, the rates of insurance increase INSURANCE. 245 with the danger to which the property is exposed, or else the company secures itself by inserting in the policy ex- ceptional matter called the war clause. The rates of insurance vary according to the exposure of the property and the character of the property itself; the greater the risk, the higher the rate. Insurance can be obtained from one day to a term of years, giving a range of rates, from a small fraction of one per cent, to three, four and even higher rates per cent. 1. How much should be paid to insure a house valued at $1674, premium being IT%, and policy $1.50? 2. At 2^% premium, what should I pay on $6710 worth of goods? 3. At 4% premium, what should I pay on machinery and material in a factory, the estimated value of which is $6600? 4. A company takes a risk of $35000 in a block of buildings, at l-g-%, and re-insures $15000 in another com- pany at l|r%; how much premium does it realize? 5. At y 1 ^ of \% for 10 days, what should I pay on $30000? 6. I have insured $16000 for 3 months at T 5 a of \%\ how much should I pay? 7. What will be the insurance on merchandise worth 675, to be shipped from Liverpool to New York, at 4: guineas per cent.? Answers: 28 7s., $26.61, $167.75, 30.5, $80, $30, $264.00, $337.50. 246 NELSON'S COMMON-SCHOOL ARITHMETIC. XXXIII. DUODECIMALS. 274. DUODECIMALS, like decimals, is a species of cal- culation which enables the operator to compute fractional quantities as whole numbers. 12"" fourths make 1 third. 12'" thirds make 1 second. 12" seconds make 1 prime, or inch. 12' primes or inches make 1 foot. 1 inch is the fa of a foot. 1 second is the j 1 ^ of an inch, or y^ of a foot. 1 third is the fa of a second, jfa-g of a foot. 1 fourth is the fa of a third, or 2 ^-^g- of a foot. As applied to mechanical pursuits, duodecimals have seldom to be subtracted or divided; hence, the following exercises will be confined to multiplication and addition exclusively. 1. Multiply 2 ft. 5 in. by 3 ft. 4 in 25 ! Writing the dimensions as in the margin, we com- 3 4 mence with the left-hand figure (3 feet), and say, 3 ^ jr times 5 inches are 15 inches, or 1 foot 3 inches; write Q Q 3 in the inches' place. 2. Then 3 times 2 are 6, and the 1 foot carried makes 808 7 feet, which we write in the place for feet. 3. We next multiply by the 4 inches; that is, we multiply 5 inches or y 5 of a foot by 4 inches, or - 4 ^ of a foot. The result will be -f-fr, but to avoid fractions, we call the result 20" or 1 inch 8". Write 8 inches to the right of and below the 3 inches. 4. Then 4 times 2 are 8', and !'=$' or inches, which we write in the inches' place. 6. We now proceed to add them. There being nothing to add to 8", we set it down ; then 9' and 3' are 12' or inches=l foot. Write and add 1 to the 7, makes 8 feet: the product of 2 ft. 5 in.X3 ft. 4 in.=:8 0' 8'. DUODECIMALS. 247 ANOTHER WAY. 2. Multiply the following dimensions together : 10 ft. 7 in.X3 ft. 8 in.x7 ft. 9 in. 10 .7 Here we commence to multiply by 3 8 the left-hand figure (3), and write the ~~3fi 9? result without reducing to a higher denomination. 3X10 ft=30 ft., and 7 in.X3=21 in. Then multiplying by 30 101 56 1st pro. the 8, we write the first product un- 7 9 der itself as the multiplier, and the 210 707 392 second product, 56, one place further 270 909 504 ^ ^ ie r *&kt. Adding these, we have the product of two divisors. Proceeding in the same way with the 7 and 9 of the third dimension, we add together the products and reduce them to higher denominations, by which we get 300 ft. 8' 11", or 300 J* 2 ft.+Jj1 =300J ft., nearly * ft. in. ft. in. ft. in. ft. in. ft. in. 3. 17 IX 3 4=? 7. 4 8X6 4x17 2=? 4. 14 6X 7 8=? 8. 3 9X2 6x11 0=? 5. 21 9X14 11-=? 9. 21 11X6 7x17 8=? 6. 18 8X16 7= Total answers : 802 1' 3" and 3159 6' 3" 8'". ft. in. ft. in. ft. in. ft. in. ft. in. 10. 21 7XH 10=-? 13. 31 7X3 2x3 3==? 11. 13 9x17 4=? 14. 26 3x9 5x7 7=? 12. 33 7X29 3=? 15. 17 1x6 7x0 9=? Total answers: 1476 7", 2283 10' 9" 6'". Mechanics preferring common or decimal fractions to duodecimals, seldom use the latter. The following example is worked by both methods: *For this very simple method, we are indebted to J, C. Kinney, Esq., of Reading, Ohio, not having seen it before. Its simplicity would suggest it as the best method to teach this otherwise difficult rule. 248 NELSON'S COMMON-SCHOOL ARITHMETIC. 16. How many squares of flooring in 3 rooms measuring 18 ft. G in.Xby 15 ft. 8 in., and what is the cost of lay- ing, at 50 cents per square? -X^Xi=869 ft., or 8.695 sqs. 18 6 15 8 277 6 12 4 289 10 3 869 6, or 869J sq. ft., which, reduced to squares of 100 feetr=8.695 squares. 8.695x50 cents 4.347, or $4.35. 17. What is the cost of laying 4 floors of the following dimensions, at 75 cents per square? 18 ft. 9 in.Xl7 ft. 3 in. 18. What will be the cost of shingling a roof which measures 53 ft. 6 in. long, and 5 ft. 8. in. from the ridge to the outer edge of the wall, at $1.50 per square? 19. The average breadth of a board is 1 ft. 4 in., and the length 23 ft. 9 in.; what number of feet does it con- tain? 20. How many solid feet in a log measuring as follows? 45 ft. 4 in.Xl ft. 6 in.Xl ft- 3 in. Answers: $9.09, 85 ft., $9.70, 31| sq. ft. XXXIV. INVOLUTION EVOLUTION. 275. THE process of multiplying a number by itself a certain number of times is called Involution, while that of finding the number thus raised, or the reverse process, is called Evolution. IN VOLUTION EVOLUTION. 949 276. A number multiplied upon itself is raised to the power; the second power multiplied by the number is raised to the third power; the number of the power be- ing indicated by the number of times the original number has been used. The second power is also called the square, because the number of square feet, inches, etc., is found by multiply- ing the number contained in one side by itself. For a similar reason the third power is called the cube. 277. The power of a number is indicated by a small figure over the right of the number, thus : 5 3 , which shows that the third power of 5 is understood. This figure is called the index or exponent. 278. Decimals are raised to any power in the same way as whole numbers, with the difference of placing the deci- mal point, while common fractions are involved by multi- plying the numerators and denominators separately. The second power of .5 is .5X-5 or -25, and the second power offis|x|= ? V 279- The product of two numbers can not consist of more figures than there are in the two factors, and can consist of only one less than in the two factors. Take 9, the largest number of a single figure, and multiplied by itself, it produces only two figures, 81 ; and take 10, the small- est number of two figures, and multiply it by itself, and it produces three figures. This principle applies in find- ing the roots of numbers. 1. Square the following numbers: 3, 7, 9, 4, 6, 15, 27, 89, 97, 112. 2. Raise the following numbers to the powers indicated : 3 s 9 5 , 26*, 30 5 , 87*, 250 3 , 189 3 . 280. The number from which any power is raised is called -the root of that power, and the process of finding that number is called extracting the root. 250 NELSON'S COMMON-SCHOOL ARITHMETIC. 281. The root of a number derives its name from the exponent of the power, the second or square root being from the second power, the third or cube root from the third power; and is indicated thus: -j/, square root; ^K, cube or third root; j^, the fourth root. 282. A complete power is one which can have its root extracted. A surd is one which can not have its root extracted. 4 is a complete power, while 5 is a surd. TABLE. THE SQUARE ROOT OP 1=1 36= 6 121=11 256=16 4=2 49= 7 144=12 289=17 9=3 64= 8 169=13 324=18 16=4 81= 9 196=14 361=19 25=5 100=10 225=15 400=20 283. Since the product of any two numbers can not consist of more than four or less than three digits or fig- ures, the number of which a root is composed can be found by separating the squares into periods of two figures each, thus : 256 indicates that two figures formed the root, 2,56 being the periods, the root of which is 16; and in the same way 746496 is pointed, 74,64,96, indicating that three figures composed the root. The square root is 864. THE EXTRACTION OF THE SQUARE ROOT. 1. The square root of 765625 is how much? 76,56,25(875 64 1. Commencing at the right-hand, the power is 17 M 9"R separated into periods of two. 7 1 TfiQ ^* ^ e nearest S( l uare root of the last period is then taken, which gives 8, or 800. [Art. 83.] 1745 )8725 Writing the 8 in the quotient, and squaring the 8725 number, we have 64 (6400), which is written un- der the 76, or 76,00,00 INVOLUTION EVOLUTION. 251 3. Subtracting this 64 from the 76, we have a remainder of 12, to which another period (56) is annexed, making 1256. 4. For a part of the new divisor, the 8 of the quotient is doubled, giving 16 as a trial divisor. Finding it is contained 7 times, (the 7 being included in the divisor,) that figure is annexed, making 167, and the product is completed. 5. 1169 being subtracted from 1256, leaves 87. 6. Annexing the last two figures of the dividend, the last figure of the divisor is doubled, as before, making 174. This number, with the last figure of the quotient (5) annexed, is contained in 8725, 5 times without a remainder, making the square root 875. RECAPITULATION. Separating the power into periods, we find the highest root of the last period. This we place as the first quotient figure, and subtract its square from the period. To the remainder annex the next period, and for a trial divisor double the last figure of the divisor. To this divisor we annex the next quotient figure and multiply as in long division. To the next remainder is an- nexed the next period, and to the last divisor is added its last figure, which is the same as to double the quotient, and the opera- tion proceeds as before. 2. -t/1683129 is how much? 1,68,31,29(1297.354 22 2 249 9 2587 7 25943~ 3 259465 5 2594704 68 44 2431 2241 19029 18109 92000 77829 1417100 1297325 NOTE 1. This answer may be carried out to any number of places by annexing ciphers, as has been done to produce the .354 of the quo- tient. Three figures, however, are sufficiently correct for practical purposes. 2. To find the square root of a decimal quantity, we commence at the left to point off the periods of two figures. 146.739 would be pointed thus: 1,46.73,90. 11977500 10378816 252 NELSON'S COMMON-SCHOOL ARITHMETIC. The pupil can prove the accuracy of his calculations by squaring the root obtained. 3. v/14161. 8. j/16820.17. 4. 1/625. 9. 1/23467.809. 5. !/ 99980001. 10. ^167037^827 6. j/99999.8000001. 11. T /456789.375. 7. y/7837619. 12. |/10963.849. 284. The square root of a fractional number is found by extracting the root of each term. The square root of sV : =V / 2 4 7= 1- 285. Decimals are pointed off in periods from the right. .31671 is pointed thus: .31,67,10. 286. The square root of the product of two numbers gives a mean proportional between them. |/5x20= 1/10010, the mean proportional between 5 and 20. 287. The square root of the area of a square is equal to the length of the side.* 13 to 17. Find the mean proportional between 7 and 175, 121 and 36, 6 and 24, 42 and 38, 16 and 49. Answers: 35, 28, 66, 39.949+, 12. 18. A square garden contains 2916 yards; what is the length of a side in feet? 19. A pavement is 112 feet long and 7 feet broad; what will be the length of the side of a square of equal area? 20. How many yards of ground in the side of a square which would be equal to a lot measuring 144x196? 21. What is the length of the side of a square piece of land which contains 25600 acres? Answers: 28, 168, 162, 160, 2023.85+. *The area is the contents of the surface, or the number obtained by squaring the side of a square. INVOLUTION EVOLUTION. 253 288* The surfaces of circles are to each other as the squares of their diameters or circumferences. 289. A triangle is a figure having three sides and three angles, or corners. If one of these angles is square, it is called right angle and the triangle is called a right-angled triangle. 290. Two square figures, one having each of its sides equal to the perpendic- ular, and the other having each of its sides equal to the base, will, together, be equal to a square, each of whose sides is the same length of the hypotheuuse.* Let the perpendicular be 3, the base 4 and the hypoth- enuse 5. Then 3X3= 9, the area of the first square. 4X4=16, " " " " second " 25, " " " " third " But 25 is the hypothenuse squared or multiplied by it- self; therefore the square root of the square 25 will be the length of the hypothenuse. The square root of 25=5. Hence, the square root of the sum of the squares of the base and perpendicular will give the hypothenuse. And the square root of the difference ,of the squares of the hypothenuse and either of the two sides will give the third side. 22. What length of a ladder will reach across a 15-foot alley to the top of a house 30 feet high? 1125, the square root of which is 33 ft. 6J t?i., nearly. *The pupil should construct a diagram, with these squares his slate. 254 NELSON'S COMMON-SCHOOL ARITHMETIC. 23. What is the diagonal of a room 18 feet by 16? 24. A ladder 30 ft. long, placed between two trees, reaches to the height of 27 feet on one of them and 25 on the other; what is the distance between them? XXXV. EXTRACTION OF THE CUBE ROOT. 291. THE cube root of a number is such a number which, if multiplied upon its square, will make that num- ber : 2 2 X2, or 2 3 r=8. The cube root of 8=2. The sign of the cube root is -j*K. The cube root of any number consisting of three figures will be a number represented by one figure ; the cube root of a number containing more than three and less than seven figures, will be one containing two figures ; hence, we point off the figures by threes instead of twos, as iu square root. N 1. Find the cube root of 262144. 262 144(64 EXPLANATION. The near- g3 216 est cube root of the first be- 10800 ' placed to the left and its 6 x30- 79? cube taken ' which is 216 P X*A^ (216000). This subtracted . _ from the dividend, leaves 11536 46144 46, to which is annexed the next period (144), making 46144. For a trial divisor, 6 3 is then multiplied by 300, giving 10800, which is contained in 46144, 4 times. The part of the root previously obtained is multiplied by this, and that product by 30, giving 720. The square of the last, figure of the root is then taken, and the three results added, making 11536, which, multiplied by the last figure in the quotient, gives 46144. EXTRACTION OF CUBE ROOT. 255 2. Find the cube root of 17596287801. 17 V 596 V 287 5 801(2601 2*= 8 2 2 X300 =1200 )9596 2 X<>X30= 360 6 2 = 36 1596 9576 20287801 260 2 X300 =20280000 260X1X30= 7800 I 2 = _1 20287801 20287801 This differs from the last example only in the cipher of the quo- tient. Finding the trial divisor (26 2 X300) was not contained in the new dividend, a cipher was annexed to the quotient, and another to the trial divisor, giving 260 2 X300 or 20280000. This being contained in the dividend 1 time, the former part of the quotient was multiplied by it and 30, and with the square of the last figure (1) added to the trial divisor, as before, giving 20287801, which, multiplied by 1, completed the extraction. The cube root is 2601. RULE. Find the greatest root of the left period, place it in the quo- tient and divisor , and subtract its cube from the dividend. To the remainder annex the next period, and, for a trial divisor, multiply the square of the root thus obtained by 300. Divide the new dividend by this divisor, and enter the product of it, with the root already obtained and 30, under the divisor ; under this enter the square of the last quotient fyure, and the sum of the three numbers will be the true divisor. Multiply this divisor by the last quotient fyure, and s?/7>- tract the product from the dividend; to the remainder annex another period, and proceed as before. 25G NELSON'S COMMON-SCHOOL ARITHMETIC. 3 to 7. -^389017=? ^259696072=? ^5735339=? ^219365327791=? f 99252847= ? APPLICATION OF CUBE ROOT. 292. A cube is a solid body, having all its sides of equal length. Any two sides of a cube multiplied to- gether will give the superficial contents of one of the faces of the cube, and this multiplied by another side, will give the solid contents; therefore, the cube root of the number of feet, yards, etc.. contained iu any solid, will give the side of a cube of equal bulk. 7. An irregular block of stone contains 15781 cubic feet and 1333 cubic inches, or 27270901 cubic inches; what will be the side of a cube of equal solidity? 8. Required the depth of a cubic cistern that will con- tain 3375 feet. 9. What will be the side of a cubic bin or box that will contain 20 bushels of wheat? Answers: 15 ft.; 35.03 in.; 301 in. or 25 ft. 1 in XXXVI. AEITHMETIC APPLIED TO THE TRADES, FARMING, ETC. CARPENTRY. THE Carpenter proper may be called the outside and the joiner the inside carpenter. The distinction, is seldom ob- served. Master Carpenters are sometimes called Builders. They will contract for the entire work of an edifice, and super- intend its construction. The legitimate business of the carpenter is to prepare and fit all the wood-work used in CARPENTRY. 257 building houses. His prices depend on the quality of ma- terial and style of finish. Plain work on one side of white pine lumber is taken as the unit of measurement, and is called 1 Plain work on poplar is 1J Plain work on ash, oak, etc 2.} Plain work on maple 3 Segmentat or Norman work on white pine 2J Gothic work on white pine 3 Serpentine, or the Oriental variety, plain 5 Domes 9 Floors, roofs, partitions and weatherboarding are meas- ured and charged for by the square 100 sq. ft. The quantity of good lumber required for a square of flooring is 112 feet, -J- or 12 J% being allowed for waste. The quantity of good pine shingles required for a square of roofing is 1000. USEFUL HINTS ON BUILDING. Persons about to erect buildings in cities, should obtain a permit from the Board of Improvement, else they subject themselves to damages for placing obstructions in the street. Proprietors of adjoining lots and buildings should be notified of the digging of cellars, or the making of other excavations that would endanger their property. In Cincinnati, cellars may be dug 12 feet deep without, the risk of incurring damage. To excavate or build cellars deeper than this, or to build sub-cellars, permission should be obtained from adjoining proprietors. Where property is valuable, it will be to the advantage of persons who purpose building, to remember that vaults, cellars, cisterns, etc., may be said to occupy no room; and that license may be obtained from the Board of Improvement to extend cellars or vaults under the sidewalk or pavement. Foundation walls should be made thicker than the main walls of buildings, and the latter, to be secure, should rest on the middle of the foundation, allowing it to project on both sides. 258 NELSON'S COMMON-SCHOOL ARITHMETIC. PRICES OF CARPENTER WORK, WITH C10LU AT PAR, FOR A DWELLING-HOUSE OF TEN ROOMS. Cellar windows, usual size, with sash $1.50 Cellar steps, good 2.00 " doors, each ,... 4.00 First floor joists, IJin., per square,, , 0.75 Floors, per square 0.65 Trimmers, per foot 0.10 pinning, " " 0.02 Roofing, per square 1.50 Hip rafte*, per foot 0.06 Valley " " " 0.06 Cants 0.01 " 0.03 Trap-doors, each 1.50 Ceiling joists, per square... 0.50 Partitions, " ... 0.35 Poor heads, " ... 0.35 Inside door-frames, per ft.. 0,04 Outside " " ".. 0.05 Window frames, plain 0.04 Beads for do., soft wood.... 0.01 " " '* hard " .... 0.01 J Box. W. frames, soft 0.08 " hard 0.11 " " beads 0.01 J R. sills, ,.. 0.06 M. rails..,.! 0.06 Betting frames .,,,,.,.,.,. 0.07 Pocket and pullies, perpr.. 0.35 Hanging sashes, per pair.. 0.20 Base, 6 in. wide, plain, per lineal foot ,..,.. 0,05 JJase mold..,..,.... 0.08 << " large. ,. 0.09 Casings, 6 in 0.05 " 5 ..,., 0.04 Plinths, each., , 0.10 Caps, 7 in., per foot ,0.10 " 8 " '< " 0.08 " 10 " M 0.10 Cornice 0.25 ^helves, conimon,.,.,. , 0.03 Shelves, best $0.05 Cloak rails. 0.03 Trimming doors 0.30 Door sills 0.20 Mantels, each 5.00 " common, each...< ( 4.00 Cupboard front, per sq. ft.. 0.04 Window stools, per foot 0.06 " " 0.18 " large 0.22 Cornices. Gutters on eave inverted, per lineal foot 0.13 Do., mold 0.18 Do., 3 members 0.23 Do., 4 " 0.25 Do., mold 0.37 Do., " brackets 0.40 Do., 6 members 0.30 Do,, 6 modillions, etc 0.42 Do., truss, each $3 to 5.00 Truss usually referred to measurer. Fastening ornaments 0.04 Bracket cornices 0.05 Lining from $0,04 to 0.05 Tubes 0.25 Porticoes. Square colums, 8 to 9 in., per foot 0.15 Capitals, each 0.75 Best 1.25 Porch, front, per foot 0.15 " panel 0.20 " extra 0.35 '< cornice, plain 0.30 " full..,,,, 0.50 Ornamental left to meas- urer. Floors and roofs, per sq. ft. 0.25 Framework 0.20 Sills O.tt MASONS' WORK. 259 In carpenter work, the cost of material is about equal to the cost of labor. For calculations, see duodecimals, page 246. MASONS' WORK. 293. A bill of prices and a standard of measurement are generally fixed upon by mechanics of the various cities of the Union, and contract work is charged for at a cer- tain rate per cent, on the bill, according to agreement. Master masons, carpenters, etc., usually select one of their number who is expert in figures, and a good judge of work, to attend to the measurement of work done. This person is called a measurer, and his fee is paid one- half by the workmen and the other by the employer. The following rules are taken from the Cincinnati Stone-ma- sons' bill of prices: "RULES FOR MEASURING. "1. All work is to be measured by the perch of 24J cubic feet. " 2. All work to be measured from each outside corner, including all openings under eight feet wide. "3. All openings less than five feet wide to measure solid, and round their jambs, provided there is no frame; but if there is a frame, they measure solid and half round their jambs. "4. Chimney abutments and common pillars to meas- ure front and both ends for length. "5. All walls, however thin, to be reckoned eighteen inches thick. " 6. All partition walls to be measured from out to out."* * "NOTE. The above rules are for workmanship only. When the 260 NELSON'S COMMON-SCHOOL ARITHMETIC. When measuring hewn stone, neatly piled before it is laid in the wall, and in computing masonry for public works, it is customary to reckon 25 feet to the perch. Twenty per cent, is deducted for loose stone in piles. 1. How many perches in a pile of hewn stone meas- uring 28 feet long, 5 feet high and 4 feet broad? 28X5X4 - --- =22^<>, or 22 per. 10 ft. 2. What quantity of stone is in a pile 150x15x12 feet? 3. What will it cost to build the foundation of a house, which is 75 feet long, 16 feet wide and 7 feet high, wall 18 inches thick, @ $2.25 a. perch, including materials? 4. What will it cost to build a cellar that is 16 feet square, with walls 2 fe,et thick and 8 feet high, @ 82.75 per perch, material included? Answers: $173.73. 1080 perches, $113.77. BRICKLAYERS' WORK. 294. Bricklayers' work is computed by the thousand bricks. The usual dimensions of a brick are 8 inches long, 4 inches broad and 2 inches thick. There are 21 bricks in a cubic foot of wall, mortar included. The gov- ernment standard is 1000 bricks to 40 cubic feet. A brick of the above dimensions weighs 4^ Ibs. A bushel of sand weighs 113 pouuds ; of slacked lime, 51 pounds. workman furnishes materials, the value of the materials, both stone and mortar, in the above extra measurements, as well as the value of the materials saved by doors and windows, is to be deducted from the foot of the bill; that is, neither stone nor mortar is to be charged in any instance where they are not used." BRICKLAYERS' WORK. 261 RULES OF MEASUREMENT OF THE BRICK- LAYERS OF CINCINNATI. "All lengths shall be exterior or taken on the outside, from corner to corner, and for every return or cross sec- tion at openings deducted, nine inches by height for work and materials, or one foot by height for workmanship shall be allowed. Autaes and Pilasters returns allowed in all cases. " All octagon or circular work of a radius of three feefc or less shall measure double, and for larger radius a less but fair allowance shall be made. All walls cut up for or coped with brick, shall measure one foot additional height, and all walls both cut up for and coped with brick, two feet additional height. All work with a batter or mate- rial deviation from plumb line, shall measure once and a half. "Flemish or plumb bond fronts shall measure solid in all cases, and stock or pressed brick with tuck joints shall measure double, with the additional cost of pressed brick if furnished by the contractor. " Kettles or stills shall measure solid, and those of 50 feet or less, exterior surface, once and a half; and those of 25 feet or less, exterior surface, double. Fire fronts, when cased, double measurement. "Culverts or sewers, 9 inches thick and two feet or less i in diameter, and those of half-brick thick and three feet or less in diameter, shall measure solid. All circular work shall measure exterior girth. "All openings of sectional area greater than ten feet ;ii shall be deducted, except those in bond fronts and ovens, which shall not till exceeding fifty feet; then half of such ; | shall be deducted. 362 NELSON'S COMMON-SCHOOL ARITHMETIC. "Twenty-one bricks shall be allowed to the cubic foot for all brick-work, and the same proportion for thicker or thinner walls from one brick in thickness upward; forty- three bricks shall be allowed per yard for brick paving when flat, and eighty for paving on edge in ascer- taining the amount of bricks only; but when paving is done and measured as brick-work, the sand shall be al- lowed for in the measurement, and all materials on edge or cut to, waste allowed for also. "Cisterns, when measured for brick-work and materials, half brick wall shall count six inches and whole brick wall ten inches, and cistern arches and those over circular vaults shall in all cases measure double. "The floor joists shall govern the height of stories in all cases till two stories make more than twenty-four feet for carrying materials, and then twenty-four feet shall be allowed for two stories and each ten feet of the additional height** a story. When new work is built upon an old building, or one built by another contractor, all such work shall be measured and allowed for an additional story in height for labor of carrying materials. "Old bricks in piles shall be subject to the usual dis- count in ascertaining the quantity, and in the absence of special agreement shall, if on the premises and sound, be valued same as new bricks at the kiln; if from a burnt building, or otherwise unsound', shall be valued by the measurer accordingly. "When hands or materials are furnished for an em- ployer, in the absence of agreement or contract, actual cost, together with fifteen per cent, thereon, shall be charged by the contractor. "Lumber and materials for scaffolding, mortar beds, etc., and vessels for holding water, shall, in all cases, be fur- nished by the employer, unless otherwise agreed upon, iu STONE-CUTTERS. 263 which case a reasonable charge shall be made by the con* tractor. "For all work not embraced in, or provided for by the foregoing, a fair and reasonable allowance shall be made by the measurer." 1. Plow many thousands of brick will be required to build a wall 90 feet long, 6 feet high and 20 inches thick? 90X6=540 square feet of surface. In one square foot of a 20-inch wall there are 35 bricks j in 540 square feet there are 540x35 = 18900 bricks, which, divided by 1000=18.9, or 18 T 9 . 2. In a house there are 6200 square feet of 20-inch wall and 2000 square feet of 12-inch wall; what will be the cost of building, at 11.50 a thousand, including price of brick and laying? 3. What will it cost to pave a yard 25 by 50 feet, and a walk 75 by 5 feet, @ 50 cents a yard including mate- rials ? Answers: $90.28, 2978.50. 4. A cistern is 8 feet in diameter and 12 feet deep (av- erage measure) ; what will be the cost of building at 40 cents a barrel? 8 2 X12X.1865=143.23 bbls., 143.23x40c.=$57.29. Instead of multiplying by .7854 (as required by Art. ), and dividing by 4211 (the number of feet in a bar- rel), we merely use the quotient arising from .7854-7- 4.211:=r.l865, as a multiplier. The true pitch of a roof is obtained by making the rafters three- fourths of the width of the building. The Gothic pitch is that produced by making the rafters as long as the building is wide. For information relating to the department of Building, we are indebted to R. B. Moore, Esq. 264 NELSON'S COMMON-SCHOOL ARITHMETIC. STONE CUTTERS. AVERAGE PRICE AND RULES OF MEASUREMENT FOR COM- MON FREESTONE WORK. 295. In measuring plain stone work, all the dressed faces of the stone are taken, and the whole reduced to su- perficial measurement.- For instance, a step 4 feet long, 14 inches wide and 7 inches thick would be measured as follows : Length, 4 ft.-f 7 in. -(-7 in.=r:5 ft. 2 in. Width, 14 +7 =1 9, and 5 ft. 2 in.Xl ft. 9 in.=9 ft. 1", or 9 ft. REMARK. It will be observed that the ends have been measured twice; this is in accordance with custom. Window-sills are measured by the running foot, includ- ing the projections of the ends. Prices, for 7-inch wide, and 4 to 5 thick, per foot, 18 to 25 cents. Water-table. The stone in front of a house and on a level with the door-step. Measured as above, 37-^ cents per foot. Ashler or slab front. Face measure, adding all worked ends. Price, 40 cents per superficial foot.* Flagging. Superficial measure, 2^- inches thick, 25 cents; 3 inches, 30 cents; 4 inches, 40 cents; 6 inches, 50 cents; 8 inches, 60 cents. Fire-wall doping, running measure, 11 by 2 inches, per foot, 25 cents. Chimney coping, 2^ to 3 inches thick, per foot, 30 cents. Coping caps, common size, each, $2. Hearths, common thickness, per superficial foot, 40 cents. * Stone-cutters set (build) their own work, the charge for which is included in these prices. PLASTERING. 265 Edge curlings, for walks, 2 to 3 inches thick, per linear foot, 25 cents. Door and window cornice, not exceeding 6 inches thick and 6 inches projection, per foot, 37-J- cents. Measured length and returns 3 girts from wall to wall. Piers for open fronts, face measure, taking the girt. A common door-piece, comprising 2 each, plinths, piers and caps, with lintel, cornice and blocking, will cost from $50 to $75. All cut stone should be laid in cement. Foundations and excavations for steps should be sunk at least three feet below the surface, else the action of the frost on the earth will be liable to displace the stone. Mortar should not be exposed to the action of frost until it is set. During the heats of summer, mortar is injured by a too rapid drying in the wall; to prevent this, the other ma- terials, stone or brick, should be thoroughly moistened before being laid; and afterward, if the weather is very hot, the masonry should be kept wet until the mortar gives indications of setting. In very warm weather, the top course should always be well moistened by the work- men on quitting their work for any short period. PLASTERING. 296. The business of the plasterer is to cover brick and stone work, ceilings and partitions, with plaster, and prepare them for paper, paint, etc. ; also, to form cornices and such other decorative portions of walls and ceilings as may be executed in plaster or cement. To lay off a square corner or a right angle, with a carpenter's rule: Measure 3 feet from the corner in one direction, 4 feet in another direction and separate the points 5 feet apart. 23 266 NELSON'S COMMON-SCHOOL ARITHMETIC. "CINCINNATI PLASTERERS' RULES OF MEAS- UREMENT. " 1. All work shall be measured superficially, including openings. All heights shall be taken from the floor to the ceiling. U 2. All staircases eight feet wide and under shall be measured double; all over eight feet, once and a half. "3. All passages four feet wide and under shall be measured once and a half; all over four feet, once and a fourth. "4. All inclined ceilings to measure once and a half. "5. All dormor windows, closets and privies to be meas- ured double. " 6. All octagon and circular work, except ceilings of rooms, to be measured double. All arched ceilings of rooms to be measured once and a half. "7. The deductions for openings occasioned by doors and windows, when the workman furnishes materials, shall be, for lathwork, one-eighth; for brick walls, one-fourth. "8. The materials for scaffolding and mortar-beds and vessels for holding water, are, in all cases, to be furnished by the employer." STUCCO WORK. Moldings, cornices not over 12 inches girth, are meas- ured by the running or linear foot. Flowers^ sometimes singly; in moldings, per superficial foot. Eighteen laths will cover a yard; 500 laths, a square of 100 feet. Estimate work is made from a bill of prices, the carpenter agree- ing to work for a certain percentage on the bill. PAINTING, PAPER-HANGING AND GLAZING 267 1. "What will be the cost of plastering a room 18 feet long and 16 feet wide, with a ceiling 10 feet high, at 18 cents? 18 J.6 34x^68, length round the room. 68x10=680 square feet in walls. 18X16=288 " " " ceiling. 9)968 107.55 yds. at 18c.=$19.359, or $19.36. 2. How many square yards of plastering in a hall 84 feet long and 40 feet wide, with a ceiling 18 feet high, having a, space of 600 square feet occupied by windows? Ans. 802f yards. PAINTING, PAPER-HANGING AND GLAZING. 297. Painters' Work is measured by the square yard, and charged for according to the number of coats, the quality of the paint and the description of the work. Sash frames are charged for singly or by the piece, and sashes by the number of squares. Lettering is charged .for by the lineal foot. Common lettering, 25 cents; gild- ing, 75 cents. Painting is sometimes charged for by the quantity of paint used, and the time spent in putting it on. The calculations being so simple, it is considered un- necessary to give any examples. 298. Glazing is sometimes charged by the square foot, and sometimes so much per light. When estimated by the foot, it is usual to include the sash in the measure- ment. 299. Paper-hanging is charged for by the piece. The commoner qualities measure about 7f yards by 19 inches, 268 NELSON'S COMMON-SCHOOL ARITHMETIC. or about 35 square feet; the better qualities, say from 50 cents a piece, are 9 yards by 21 inches, or 47 square feet. Border paper is made in rolls of the same dimensions as the wall paper, each roll or piece containing two or more strips, each of which is called a piece, and is sold at about the same price as the paper it is designed to match. Dealers in paper usually contract for the hanging, and charge from 20 to 25 cents a piece, according to quality. 300. To find the quantity of paper required for a room, compute the number of square feet in the walls; deduct the openings and divide the result by the number of feet in a piece. The space occupied by the base will allow for waste and matching pattern. 1. A room is 16 feet square and has a ceiling 12 feet high, with two doors 7 feet by 4, two windows 7 by 3 feet and a fire-place 4 feet square; how much paper will be required to hang it, and what will be the whole cost, including hanging paper 25, border 35 cents? 16 X 4=64, length of wall around the room. 64X12=768 square feet of wall Windows, 42 feet. Doors, 56 " @ Fire-place, 16 " 114 654 square feet to be covered. 654 =19 pieces. 35 number of feet in a piece. 19X25 cents=cost of paper 4.75 Length of border, 64 feet, or 3 pieces at 35 1.05 Hanging 22 pieces @ 20 cents,* 4.40 Total cost $10.20 * The border is included in the 22 pieces. FARMING. 269 GAS FITTING AND PLUMBING. 301. Gas fitting is charged for per foot of pipe, vary- ing according to size. Fittings and chandeliers, per piece. 302. Plumbing is charged for like gas fitting. For ordinary house work, say from 25 to 30 cents per foot of pipe ; sheeting by weight. FARMING. The young farmer will find it to his interest to be a good arithmetician. For those who have not had the ad- vantages of an early education, we will introduce a few of the simpler and more necessary calculations, suggesting, at the same time, that during his leisure moments the farmer should master the entire science as contained in this little work, which any person of ordinary ability, who can read and write, may accomplish without the aid of a teacher. 303. To find the number of acres in a field or tract of land having four square corners,* we multiply the length by the breadth, and divide the result by 160, if the measure was taken in rods; or by 43560, if taken in feet.f 1. The length of a field is 125 rods and its breadth 112 rods; how many acres are in it? 112X1^5=14000, which, divided by 160, gives 87J. * A figure having square corners, and all its sides equal, is a equare; one having its opposite sides equal, a rectangle or parallelo- gram. tin a square rod there are 272|- square feet. When there are feet remaining to be reduced to rods, it will be sufficiently accurate to divide by 272. 270 NELSON'S COMMON-SCHOOL ARITHMETIC. 2. A lot of land is 400 feet long by 110 feet broad; how many acres does it contain? Ans. 1 acre 1.7 rods. 304. To lay off a given quantity of land. 3. What should be the length of a strip of land 30 rods broad to contain 6 acres? In 6 acres there are 960 rods, which, divided by 30= 32 rods. 305. To find the contents of a field in the shape of ct right-angled triangle, we multiply the two shorter sides to- gether, and take one-half the product. REASON. A right-angled triangle is half a square or parallelo- gram, formed by drawing a line between opposite corners. 4. The shorter sides of a right-angled triangle are 45 and 60; required the contents. Am. 1350. 306. To find the quantity of grain or coal in a bin or wagon, we multiply the length, breadth and height to- gether, and for grain divide the product by 1.2444,* if the diminsions are given in feet; or by 2150.42, f if given in inches. For coal, by 1.555, or 2688. 5. A wagon is 8 feet long, 5 feet broad and 18 inches deep; how many bushels of corn does it contain? 8X^X1^=60, the number of cubic feet. 1.2444)60.0000(48.21, or 48 bushels.* 49776 102240 NOTE. Two ciphers were annexed to the 99552 dividend to correspond with the decimals of the divisor, and produce the ivhole numbers, 48, and two more ciphers were annexed to pro- duce the decimal .23, or - 2 75 L or i. 19920 6. How many bushels of grain in a bin measuring 4 feet every way? Ans. 51^, nearly. *Feet in a bushel. t Inches in a bushel. FARMING. 271 307. To find flie quantity of wood or bark in a pile, we multiply the three sides given in feet, as before, and di- vide by 128, the number of feet in a cord. 7. How many cords of wood in a pile 40 feet long, 7 feet high and 4 feet broad? Ans. 8|- cords. SOS. Having two sides and the contents of a box, to find the third side, we divide the cubical contents by the pro- duct of the two sides. REASON. Since the product of the three sides equals the con- tents, the contents divided by two of the sides will give the third side. 8. A box is 2 feet wide and 3 feet high; how long should it be to hold 25 bushels of coal ? In 25 bushels there are 2688X25 or 67200 cubic inches. In 2 ft. there are 24 inches. " 3 " " 36 " 24x36=864=area of the end. 672-5-864==77|5|, or 6 ft. 5f in. 9. What must be the height of a bin that will hold 300 bushels of wheat, if its- length is 30 feet and its width 4 feet? Ans. 3 ft. 1J in. 10. What must be the depth of a box 16 inches square to hold a bushel? a box 10 inches square to hold a peck? one 8 inches square to hold half a peck? To find the side of a cube that will hold a certain quan- tity. See Cube Root. 309. To find the quantity of grain when heaped against a wall or partition, take half the perpendicular height for one side, and multiply it by the length and breadth, as in Art. 306. 310. To find the number of cubic feet in a round log. See Art. on the Cylinder. To find the solidity of a cyl- inder, wa multiply the area of the end by the length. 272 NELSON'S COMMON-SCHOOL ARITHMETIC. 11. How many feet are in a log 12 feet long and 30 inches in diameter? In 30 inches there are 2J, or 2.5 feet: 2.5X2.5X-7S54 =4.9087, the area of the end. 4.9087X12=58.9044, or 58 T 9 Q feet, the solid contents. REMARK. This method of calculating, though correct, is seldom used for practical purposes. It is customary for lumber merchants to throw off one-third of the diameter, and consider the remainder the side of a square log. A log of the dimensions named in the preceding question would thus measure only 33J feet, or one-third of 100 feet; and is thereby taken as the standard of measurement in some of the Western States. See Lumber Business. 311. Trade, or barter. 12. How many cords of wood, at S3. 75 a cord, should I get for 50 bushels of wheat at $1.12J a bushel? 50x1-12^=^56.25, which, divided by $3.75, will give the number of cords. 5625-i-375 15 cords. PROOF. 15 cords at $3.75=856.25. 13. How many pounds of sugar, at 8 cents a pound, should I get for 127 pounds of buttor, at 12J cents a pound? Ans. 198. 14. How many days' work of a man, at 75 cents a day, will be equal to 45 days' work of a man at $1.25? Ans. 75. 15. How many cords of wood, at $2.25, will be equal to 150 cords, at $3.50? 16. How many yards of muslin, at 8 cents a yard, can be bought for 5 dozen chickens, at $1.25, and 15 dozen eggs, at 8J- cents? LUMBER BUSINESS. 312. Lumber measure comprises solid and superficial measure. Round logs are measured by deducting one-third LUMBER MEASURE. 273 of the diameter for waste, and calling the remainder the side of a square log. 1. To find the contents of a round log 24 inches in di* ameter and 30 feet in length. SOLUTION. Deducting J from 24 for waste, we have 16, which, squared=256 inches, and multiplied by the length^ 640 feet board measure. In some places only J is deducted for pine lumber.* Planks or joists are sometimes reckoned by face meas- ure; that is, the dimensions of one side of the board are taken instead of the solid contents. A 16-foot board 2 inches thick by 12 inches broad would measure 32 feet board measure, or 16 feet face measure. In some places, the saw-log is taken as a standard of measurement for round timber. A log 12 feet long and 30 inches in diameter is the standard in some parts of the west. In Pennsylvania, a saw-log is one that will cut into 200 feet of lumber. 313. To measure timber partly squared, it is customary to deduct the "wane" (the length of the corner) from the thickness of the log, and call the remainder one side. A log 18 inches thick, with a "wane" 3 inches, would be called one of 18 by 15 inches. 2. In an octagonal log, 25 feet long 20 inches thick, with a wane 4 inches, how many solid feet are there? Ans. 55|. 3. There are 150 logs, the average length and breadth of which are 20 feet by 22 inches, wane 3 inches ; required the number of solid feet they contain. Ans. 8708J. * Inch measure is taken as the standard for lumber. If a board is under an inch, it is measured as a full inch; and if over an inch, it is reduced to inch measurement. A plank 2 inches thick would be considered as two boards 1 inch thick. 274 NELSON'S COMMON-SCHOOL ARITHMETIC. 4. In a raft there are 450 boards 16 feet long and 1^ inches thick, and measuring in the aggregate 757 feet broad; how many feet of lumber (board measure) does it contain? How many face measure? Ans. 18168 board measure, 12112 face measure. 5. How much lumber can be cut from a tree measuring 20 feet long and 14 inches diameter at the smaller end, allowing for waste one-fourth of the diameter? HOUSEKEEPING. 314. Housekeepers, and ladies generally, ought to be familiar with the operations in arithmetic which apply in computing house rent, servants' wages, board bills, inter- est, the quantity of carpet to cover a floor or paper for a room, etc. HOUSE RENT. Landlords, in renting by the year, usually collect their rents quarterly ; but when renting monthly, collect monthly. By a quarter is meant three calendar months. As, for instance, if a house is rented on the 17th of April, the quarter would expire on the 17th of July. When a house is rented for a year, the tenant is liable for the rent during the whole of that time unless the landlord accepts another in his stead. A verbal lease for a year is binding. A lease for three or more years should be recorded. Tenancy begins on obtaining possession. When there is a lease, however, and the time not stated, it is pre- sumed to commence on the date of the instrument. When the tenant does not remove at the end of the year, or two weeks afterward, he will be regarded as hav- ing rented for another year. Interest can be collected on rent from the day it is due. HOUSEKEEPING. 275 A tenant is released when the landlord accepts a sub- stitute. A married" woman can not make a lease or take one in her own name. A tenant at will is liable for rent as long as he occu- pies the premises. For interest calculations, see page 143. 1. A house which rents at $75 a month is occupied from January 3 to February 9 ; what is the amount of rent? Rent for 1 month= 75 , " " 6dsor$= 15 $90 2. Required the rent for a house from April 3 to Au- gust 5, at $1000 a year. 3 mo= 1 1000 EXPLANATION. From April 8 to July 3 ~7^:T is 3 months, or 1 fourth of a year; 1 fourth of $1000 gives $250. From July 3 to Au- gust 5 is 1 month and 2 days; 30 days is 1 third of 3 months, and the rent for that 338.888 time is $83.333, and 2 days is 1 fifteenth or $338.89 of a month, giving the rent for that time, $5.555. 3. Required the rent for a house from December 1 to January 12, at $50 a month. Ans. $68.33. 4. What will be the rent of a house from January 20 to August 9, at $750 a year, payable quarterly? Ans. $416.67. The Teacher can give more of such questions as he finds it nec- essary. SERVANTS' WAGES. Servants are hired by the week or month of four weeks or calendar month, and are entitled to wages every day, Sunday included. 30 = 276 NELSON'S COMMON-SCHOOL ARITHMETIC. * 5. A girl hires on September 3 and leaves on October 9; what will be her wages at $3 a week? From September 3 to October 9 is 36 days, or 5 weeks 1 day. Wages for 5 weeks at $3= 15.00 Wages for I day=\ of $3= .428 15.428 or $15.43 6. A man is hired on June 9 at $40 a calendar month, and is discharged on September 3; what is the amount of his wages? Ans. $113.33. This is computed in the same way as house rent. 7. What will be the wages of a man for 7 months and 7 days at $33 a month? Ans. $238.70. To find the quantity of carpet for a floor. Most carpeting is made one yard in breadth. Brussels and velvet carpeting is usually made only f of a yard or 27 inches, though sometimes it is made |- and even f, or double the usual breadth. Oil cloths vary in breadth from 3 to 24 feet, as follows : 3 ft. 9 in., 4 ft. 6 in., 7 ft. 6 in., 12 ft., 18 ft. and 24 ft. Matting is of three kinds: China Cocoa, Manilla and Cane. China matting is made of a kind of rushes. It- looks neat but does not wear long. The best kinds are Gowqua and Manning. Cocoa matting is made of a kind of grass. The best quality is called "diamond A," from the brand found upon it. Common ingrain carpeting may be matched by cutting through the center of the pattern ; but expensive carpets can be matched only by persons experienced in the busi ness. Some of them require two webs, others more, to HOUSEKEEPING. 277 make a pattern. Carpet dealers usually furnish their car- pets made to any dimensions, and even lay them when re- quired. The quantity of carpet required for a room is found by multiplying the length by the breadth, in feet or inches, and dividing by the number of square feet or inches in a yard. For f carpet, divide the square feet by 6-J; for yard, divide by 9.* ANOTHER WAY Is to find the number of Ireadths required, and multiply it by the length of the room. 8. How much ingrain carpet will be required to cover a room 15 by 20 feet? Cutting the carpet in its greatest length, there would be 5 breadths, which, multiplied by the length, gives 100 feet, or 33J yards. Required the quantity of ingrain carpet to cover three rooms, measuring as follows: one room 12 by 16 feet; one, 16 by 21; one, 15 by 19; and one room, 20 by 25 with velvet carpet. 9. What quantity of velvet carpet will cover a saloon 20 by 40 feet? The breadth, 20 feet, reduced to inches 240, which, divided by 27 inches=9 breadths, nearly. The length, 40 feet, multiplied by 9^360 feet, or 120 yards. for calculations pertaining to wall paper, see Paper Hanging, page 267. For calculations pertaining to shopping, see page 111. For weights and measures, see Tables. * The spaces for fire-places, etc., will allow sufficient for waste. Carpets should be cut a few inches short to allow for stretching. 278 NELSON'S COMMON-SCHOOL ARITHMETIC. To find the cost of articles sold by the dozen. 1 article will cost y 1 ^- of the cost of a dozen. 2 articles " " " " " " " g u it it l u u it it it 4 u it a ^_ it it it it it 5 " " " 5 times the twelfth. 6 u " " | of the cost of a dozen. 8 " " " 2 times the third. 9 " " " 3 " " fourth. 10 " " " $ off. 11 " " " -J. " XXXVII. GEOMETRICAL DEFINITIONS. An Angle is the opening between two lines that meet in a point. A Right Angle is made by one straight line standing perpendicular to another. An Obtuse Angle is wider than a right angle. An Acute Angle is less than a right angle. A Triangle is a figure having three sides and three angles. An Equilateral Triangle has all its sides equal. An Isoscetes Triangle has two of its sides equal. A Scalene Triangle has all its sides unequal. A Right-Angled Triangle has one right angle. An Obtuse-Angled Triangle has one obtuse angle. An Acute-Angled Triangle has all its angles acute. A Quadrangle or Quadrilateral is a four-sided figure, and may be A Parallellogram, having its opposite sides parallel ; A Rectangle, having four right angles, sides unequal; GEOMETRICAL DEFINITIONS. 279 A Square, having all its sides* equal, and its angles right angles; A Rhombus or Lozenge, having its sides equal and no right angle; A Rhomboid, a parallelogram, with no right angles ; A Trapezium, having unequal sides; A Trapezoid, having only two sides parallel. Polygon, a plain figure having more than four sides. A Pentagon has nre sides, a hexagon six, a heptagon seven, an octagon eight, a nonegon nine, a decagon ten, etc. A Circle is a plain figure, bounded by a curved line, all points of which are equidistant from the center. An Arc is any part of a circumference. A Chord is a straight line joining the extremities of an arc. A Segment of a circle is a part of a circle bounded by an arc and its chord. The Radius of a circle is a line extending from the center to the circumference. A Quadrant is a quarter, a sextant a sixth of a circle. A Zone, a part of a circle included between two parallel chords. A Prism is a solid, the sides of which are parallelo- grams. It may have three or more sides. A Pyramid is a solid with regular sides, tapering to a point. A Cylinder is a solid of uniform thickness, having its ends circular. A Cone is a round pyramid, having a circle for its base. The Circumference of a circle is the line by which it is bounded. The Diameter is a line drawn through the cen- ter and terminating at the circumference. The Radius is a line drawn from the center to any point in the circum- ference. 280 NELSON'S COMMON-SCHOOL ARITHMETIC. j "1 , j. I j XXXVIII. MENSURATION. 315. To find the area of a square, multiply the length by the breadth. 1. Let one side of the annexed parallelo- gram be 3, and the other 4, then the area will be 3X4=12. 2. What is the area of a square, the side of which is 3 feet 6 inches. Ans. 12J. 316. To find the area of a Rhombus or Rhomboid, we multiply the length by the perpendicu- lar height. The reason for this will be evident from an inspection of the fig- ures. The triangle A C m of the rhom- bus applied to the side B D, will make a square ; and the triangle of the rhom- boid applied to the other side will make a parallelogram. 3. Let the perpendicular height (A m) be 20, and the length (A B or C D) be 30, then the area will be 20x30600. 4. What will be the area of a rhomboid, the perpen- dicular and length being 15 and 25? Ans. 375. 317. To find the area of a right-angled triangle, we multiply the perpendicular by half B the base, or the base by half the per- pendicular. REASON. The part A m n, applied to the j part B C m n, with the line A m applied to m B, and the point A on the point B, will make a parallelogram, with a base half of that of the triangle. MENSURATION. 281 5. Let the base be 10 and the perpendicular 8; then 1X 8=40, the area. 6. The perpendicular is 16 and the base 120; what is the area? Ans. 960. 318. When the triangle is not right-angular, half the base multiplied on the height will give the area. REASON. The triangle C A D is half the rhomboid. 7. Let the base be 30 and the height 20, then 20 X- 3 / 300, the area. 319. When the perpendicular is not given, the area can be found by subtracting each side from half of the sum of the sides; then by multiplying these three remainders and half the sum of the sides together, and extracting the square root of the product 8. Let the sides be 5, 7 and 10 ; then - - --- ^ 11 7=4 1110=1 6X4X11=264, the sq. root of which is 16.3. 9. What is the area of a triangle, the sides of which are 50, 30 and 40? Ans. 600. 320. To find the area of a circle, multiply the square of the diameter by .7854. Multiply half the circumference by half the diameter. 321. To find the side of a square equal in area to a given circle, multiply the circumference by .2820948, or the di- ameter by .8862269. 10. The diameter of a circle is 100; required the side of a square having the same area. .8862269X100=88.62269, side of a square, 24 282 NELSON'S COMMON-SCHOOL ARITHMETIC. 322. To find the side of an inscribed square, multiply the diameter by .7071068. 11. The diameter of a tree is 2 feet; re- quired the side of a square log that may be cut from it? .7071068X2=1.4142136, or 1 foot 5 inches nearly. 323. Having the side of a square, to find the diameter of a circumscribed circle, multiply the side by 1.4142136. 324. From the side of a square to find the circumference of a circumscribed circle, multiply the side by 4.4428934. 325. To find the diameter, multiply the side by 1.1283791. 326. From the side of a square, to find the circumference of a circle of equal area, multiply the side by 3.5449076. 327. To find the area of a trapezium, we divide it into two triangles, and the sum of the areas will be the area required. 12. Let the diagonal A C be 100, and the perpendiculars B m and D n : 30 and 35 ; then (35+30) X 100 V ^ ' =3250, the area. REMARK. The areas of irregular polygons are found by dividing the figures into triangles, and taking the sum of their areas. 328. To find the circumference of a circle, we multiply the diameter by 3.1416, or 3, because the circumference of a circle is 3^ times greater than the diameter. 13. Let the diameter be 5; then 3.1416 X 5=15. 708, the circumference. The diameter is found by dividing the circumference by 3.1416. MENSURATION. 283 329. To find the area of a regular polygon, add all the sides together, and multiply by the perpendicular drawn from the center of the polygon to the middle of one of its sides; or, Multiply the square of the side of the polygon by the number standing opposite to the number of its sides in the following table: 3 0.4333012 8 4.8284271 4 1. 9 6.1818242 5 1.7204774 10 7.6942088 6 2.5980762 11 9.3656404 7 3.6339124 12 11.1961524 14. The side of a pentagon (a five-sided figure) is 20 yds. and its perpendicular 13.76382; required the area. 20X5X13.76382 First method, =688.191, Ans. Second method, 20 2 X 1.720477=688.19, Ans. 15. The side of a nonegon is 50 inches; required its area. Ans. 15454.5605. 330. To measure the heights of objects, the tops of which can not be reached, the shadow cast by the tree may be used. Measure the length of the shadow cast by the ob- ject, and that of some object the length of which is known; then the shadow of the known object will be to that of the first as the length of the known object to the length of the first 16. Let the known object be a man, who, without his hat, measures (in his shoes) 5 feet 8 inches, and whose shadow measures 15 feet, while the shadow of a tree meas- ures 120 feet. 15 ; 120 : : 68 inches : the height of the tree in inches. 120X68 =544 inches, or 45J feet. 284 NELSON'S COMMON-SCHOOL ARITHMETIC. MEASUEEMENT OF SOLIDS. 331. . To find the solidity of a cube, multiply the side by itself, and the product again by the side. 17. The side of a cubical block of marble is 5 feet 7 inches; what is the solid contents? 5_7^x5 T 7 .TX5 T 7 2 =174 feet, nearly. 332. To find the solidity of a parallelopipedon (a solid figure with square corners), multiply the length, breadth and thickness together. 18. A log measures 7 feet in length and 15 by 20 inches in thickness; required the solid contents. 7X15X20=2100 square inches, or 14 T 7 ^ feet. 333. To find the solidity of a prism, multiply the area of the end by the length. 19. What is the solidity of a prism whose ends are equilateral triangles, each side of which is 4 feet and height 8 feet? An equilateral triangle is made up of two right-angled triangles, the perpendicular of which is found by taking the square root of the difference of the squares of half the base and the other side. y / 4 2 2 2 =3.464, the perpendicular, m n. 3.464x2=6.928=area of the end. 6.928X8=55.424 feet, the solidity. 334. To find the solidity of a cone, multiply the area of the base by one-third the height. 20. A cone is 10 feet in diameter and 10 feet high; re- quired the solidity. 10 2 X -7854=78.54, area of the base, which, multiplied by J the height, 3J=261.8, the solidity. MENSURATION. 285 21. How many cubic feet in a cone whose diameter is 12 feet, and its perpendicular height 100? Ans. 3769.92 ft. 335. To find the solidity of a pyramid, multiply the area of the base by one-third the height. 22. A square pyramid has a base of 4 feet and height of 12 feet; required the solidity. 4x4 area of base. 16X^=64 for the solidity. 23. The spire of a church is an octagonal pyramid, each side at the base being 5 feet 10 inches, and its perpen- dicular height 45 feet; also each side of the cavity or hollow part at the base is 4 feet 11 inches, and its per- pendicular height 41 feet ; how many solid yards of stone does the spire contain? Ans. 32|-, nearly. 336. To find the solidity of the frustrum of a cone or pyramid* Find the sum of the areas of the two ends, and of a geometrical mean between them, and multiply by one- third the perpendicular height. 24. What is the solid contents of a frustrum of a square pyramid, whose sides are 5 and 3, and perpen- dicular height 12? 5 2 X .7854=19.6350 area base. 32 X .7854= 7.0686 area upper end. . 1 /r9.635x7.0686=11.7810 geometrical mean. 38.4846 4 one- third the height. 153.9384 25. What is the solidity of a squared piece of timber, its length being 18 feet, and sides of the bases 18 and 12 inches? Ans. 28.5 ft. *A segment is a piece cut off by a plane, parallel to the base; a fruslrwn is what remains at the base. 286 NELSON'S COMMON-SCHOOL ARITHMETIC. 26. How many cubic feet of timber in a tapering log 14.25 ft. long, diameters 9 and 18 in.? Am. 14.689 ft. Comparison between the globe, cylinder and cone, the di- ameter and heights being 100: Solid contents of the cylinder, 785.4 " " " sphere, 523.6=f of the cylinder. " " " cone, 261.8=4 u " REMARKS. 1. The cone cut out of a solid cylinder, whose diame- ter and height are equal, will leave a part equal to the solidity of a sphere of the same diameter. 2. A square pyramid, whose height and side are equal to the side of a cube, if cut out of the latter, will leave f of the cube. GAUGING. The process of finding the capacity of barrels, etc., is called gauging. 337. llfiving the head and Lung diameter and the di- ameter between them, to find the capacity of a barrel or cask in gallons, we add together the square of the head and bung diameters and twice the middle diameter, and multiply the sun; by the length, and that by .0004721 for imperial gallon. 27. A cask, having for head, bung and middle diameter 30, 36 and 33, and length 40 inches, holds how many im- perial gallons? 30 2 -f36 2 -f(33x2)X40x. 0004721=42.72 galls. Practical method for measuring small cylindrical vessels, is to multiply the square of the diameter by 34, and that by the height in inches, and point off four figures. The result will be the capacity in gallons. 28. An oil-can measures 12 inches in diameter and 2 feet in height; required the contents in gallons. 12 2 X34X 24=117504=11.75 or 11$ galls. THE METRIC SYSTEM. 287 XXXIX. THE METRIC SYSTEM. 338. The Metric System is a decimal system of weights and measures, of French origin, deriving its name from Meter, the unit of measure upon which the system is based. Since 1840 it has been adopted by most European gov- ernments, including that of Great Britain in 1864, and has been in use by men of science every-where. During its last session (39th), Congress authorized its use in this country, and made provision for its immediate introduc- tion into post-offices.* 339. The units of measure are the meter, are, liter, and stere; and the unit of weight the gram. Other denomina- tions are formed from these by prefixing Greek or Latin numerals; the former for denominations above the unit, and the latter for denominations below the unit. The Greek prefixes are deka, 10; hecto, 100; kilo, 1000; and myria, 10000. A dekameter is ten times and a hectome- ter one hundred times the length of a meter. The Latin *A Bill to authorize the use of the Metric System of Weights and Measures. J3e it enacted by the Senate and House of Representatives of the United States of America, in Congress assembled, That from and after the passage of this act, it shall be lawful throughout the United States of America to employ the weights and measures of the Metric System; and no contract, or dealing, or pleading in any court, shall be deemed invalid or liable to objection because the weights or measures expressed or referred to therein are weights or measures of the Metric System. 288 NELSON'S COMMON-SCHOOL ARITHMETIC. prefixes are deci, T Vth; C entL T J-g-th; milli, y^Vu*^- A decimeter is one-tenth and a centimeter one-hundredth of the length of a meter. LIST OF NAMES AND THEIR PRONUNCIATION. Name. Pi'onunciation. Abbreviation. Meter, Me'ter, M. Kilometer, Kil'o-meter, K. M, Hectometer, Hec'to-meter, H. M, Dekameter, Dek'a-meter, D. M, Decimeter, Des'i-meter, d. m. Centimeter, Cent'i-ineter, c. m. Are, Aer, A. Hectare, Hector, H, A. Centare. Seut'iir, c. a. Stere, Stere, S. Dekastere, Deckl-stere, D. S. Decistere, Des'i-stere, d, s. Gram, Gram, G. Dekagram, Dek x a-gram, D. G. Kilogram, KiFo-gram, K. G. Hectogram, Hec^o-gram, H. G. Decigram, Des^-gram, d.g. Centigram, Sent^-gram, e.g. Milligram, Mil'li-gram, m. g. Tonneau, Ton x no, Ton. Millier, MiKli-er, Mil. Liter, L^ter, L. Kiloliter, KiKo-k-ter, K. L. Hectoliter, Ilect/o-le-ter, H. L. Dekaliter, Dek x a-te-ter, D. L. Deciliter, Des / i-le-ter, d. 1. Centiliter, SenKi-le-ter, c. 1. Milliliter, MiKli-lg-ter, m. 1. NOTE. Denominations below units arc abbreviated with small letters. THE METRIC SYSTEM. 289 MEASURES OF LENGTH. 840. Besides its being the base of the new system, the Meter is the unit of measure for lengths, and is one ten- millionth part of the distance from the Equator to the poles, and is equivalent to 39.37 inches ordinary measure. Myriameter 10,000 Meters or 393685 inches. Kilometer 1,000 " or 39368.5 " Hectometer = 100 " or 3936.85 " Dekameter 10 < or 393.685 " Meter = 1 or 39.3685 " Decimeter = T ^th " or 3.9368 " Centimeter == ^th " or .39368 " Millimeter T?r Wth " or .03936 REMARK. Meters, when combined with lower denominations^ are treated as whole numbers, and the latter as decimals. The denominations most used are the Kilometer, Centimeter^ and Millimeter. 7 Myriameters, 8 Kilometers, and 6 Hectome* ters would be written 78.5 K. M. SQUARE OR SURFACE MEASURE. 341. The unit of measure for large surfaces is the Are, from which are derived the Hectare and Centare. For smaller surfaces the denominations are the same as for measures of length, with the addition of the word square. Centare = 1 sq. meter, or 1560 sq. inches. Are = 100 sq. meters, or 1 sq. dekameter, or 119.6 sq. yds. Hectare = 10,000 sq. meters, or 1 sq. hectometer, or 2.471 acres. 1 sq. decimeter, d. w., 2 = T J^ sq. meter 1 sq. centimeter, c. m., 2 =-. T J^ sq. decimeter, or T Q JQQ sq. meter. 1 sq. millimeter, m. m., 2 = T J^ sq. centimeter, or T ^^J^^^ sq. c. m. 27.354 M. 2 = 27 sq. meters, 35 sq. decimeters, 40 sq. centimeters. 290 NELSON'S COMMON-SCHOOL ARITHMETIC. CUBIC OR SOLID MEASURE. 342. The Stere may be called the unit for cubic meas- ure. It is equal to a cubic meter or 1.308 yards. 1 dekastere, D. S.,=W steres, or 13.08 yards. 1 decistere, d. s., T V stere, or 0.1308 yards. 1 cubic decimeter, d. m.f = T ^Vtf cubic meter. 1 cubic centimeter, c. m.^= T ^ inf cubic decimeter, or -j-^J^^ M. 1 cubic millimeter, m.wi.,= T ^ cubic centimeter, or The Stere, etc., is used for measuring fire-wood and lumber, and, for computing large numbers, is preferred to the other denominations. MEASURES OF CAPACITY. 343. The Liter is the unit of measure for capacity, and is equal to a cubic decimeter or 1.0567 quarts of United States liquid measure. TABLE. Kiloliter = 1,000 liters, or 1 cubic meter of water. Hectoliter = 100 " or ^ cubic meter. Dekaliter = 10 " or 10 cubic decimeters. Liter of water weighs 1 kilogram. Deciliter = j 1 ^ liter, or ^ cubic decimeter. Centiliter = T ^ liter, or 10 cubic centimeters. A milliliter of water weighs a gram. The liter and hectoliter are most in use. WEIGHTS. , 344. The Gram is the unit of weight, and is equal to 15.432 grains Troy, which is the weight of a cubic centi- meter of pure water at its greatest density. THE METRIC SYSTEM. 291 Kilogram = 1,000 grams. Hectogram = 100 " Dekagram = 10 " Gram 1 gram. Decigram = " Centigram = ^ gram. Quintal = 100 kilograms. Tonneau 1,000 kilograms, or 2,204 Ibs. REMARK. The gram and its subdivisions are used in com- pounding medicines, and wherever great accuracy is required. The kilogram is the denomination most used, and weighs a little over 2 pounds. The quintal and tonneau are used for heavy weights, but may be expressed in kilograms. COMPARISON OF METRIC DENOMINATIONS WITH THOSE IN PRESENT USE. MEASURES OF LENGTH. NAMES. VALUES. EQUIVALENTS IN USE. Mvriameter .. 10 000 meters 6.2137 miles. Kilometer 1,0(10 ik 0.62137 " 100 " 328 l-l2th feet. Deka meter 10 " 393.7 inches. 39 37 " Decimeter Centimeter Millimeter l-10th of a meter 1 -100th of a meter 1-loooth of a meter 3.937 " 0.3937 " 0.0394 ** MEASURES OF SURFACE. NAMKS. VALUES. EQUIVALENTS IN USE. Hectare 10,000 square meter.* 2.471 acres. A re 100 Con tare 1 ki meter l">f>0 square indies. MEASURES OF CAPACITY. NAMES. No. of Liters. CUBIC MEASURE. DRY MEASURE. LIQUID OR WINK ME AS. Kiloliter or Stere. Hectoliter 1,000 100 cubic met-r -10th cubic meter 1.3()<S cubic yards... 2 bus & 3 3 r > pecks 264. 17 grails. 26.417 alls. Dekaliter 10 9 og quart* 2 6417 galls Liter 1 cubic decimeter 'MIX quart I u r >67 quarts. Deciliter Centiliter Milliliter 1-10 1-100- 1-1000 -1<> cubic decimeter. cubic centimeters cubic centimeter... 6. 1022 cubic inches. 0.6102 cubic inch ... 0.06102 cubic inch.. O.S45 trill. 0.:W8 fluid oz. 0.27 fluid drin. 292 NELSON'S COMMON-SCHOOL ARITHMETIC. WEIGHTS. NAMES. No. OF GRAMS. WEIGHT IN QUANTITY OF WATER AT MAXIMUM DENSITY. EQUIVALENTS IN USE. Millinr or Toiine.au... i 000 <H)0 1 cubic meter 2204 6 pounds Av. Quintal 1(1(1 Off! I 220.46 " 10 (too 22.046 " 1,(MXJ 1 liter 2.2046 *' Hectogram 100 1 deciliter . 3.5274 ounces Dekagram 10 10 cubic centimeters 3527 ounce 1 1 cubic centimeter 15.'432 grains T oy. 1-10 1.5432 " Centigram Milligram 1-100 1-llXJO 10 cubic millimeters 1 cubic millimeter 0.1543 grain 0.0154 " CALCULATIONS PERTAINING TO THE METRIC SYSTEM. The principal recommendation of the new system is the simplicity of its arithmetic. Under it the long and in- tricate calculations of the present system are unknown. Denominations are changed from one to another by the use of the decimal point, with sometimes one or more ciphers, while addition and other operations are per- formed precisely as in whole numbers and decimals. 1. In 35.2 D. M. how many M? Ans. 352 M. Solution. In 1 D. M. there are 10 M., in 35.2 D. M. there are 35.2 times 10 or 352 M. 2. In 35142 millimeters how many meters? Ans. 35.142 M. Solution. 1000 m. m. make 1 M., hence we divide by 1000, which gives 35.142 M. 3. In 31475 M. how many kilometers? Ans. 31.475 K. M! 4. Reduce 371.2 D. M. to meters. Ans. 3712 M. 5. lleduce 45.67 M. to decimeters. Ans. 456.7 d. c. 6. lleduce 13456 decimeters to meters. Ans. 1345.6 M. 7. In 213.21 M. M. how many dekameters? Ans. 213210 D. M. 8. lleduce 3157.2 c. m. to meters. Ans. 31.572 M. THE METRIC SYSTEM. 293 9. A man traveled 80.5 K. M. in a day, how far would he travel in 5 days at the same rate ? Ans. 152.5 K. M. 10. A ship steams 15 K. M. in an hour, how long would it take her to go a distance of 50 myriameters ? Ans. 33^ hours. 11. At 2 cents per kilometer, what would be the cost of traveling 351.27 M. M. ? Ans. $70.25. 12. A man walked a distance of 351.27 K. M. in 9 days, how many was that per day? Ans. 39.03 K. M. 13. In 253 d. m., 7 c. m., 5 m. m., how many meters ? Ans. 25.375 M. 14. In 157 d. m 2 ., 35 c. m 2 ., how many square meters ? Ans. 1.5735 M 2 . 15. At 15 cents per M 2 ., what would 25 H. M 2 . of lum- ber cost? Ans. $375. 16. How many square meters in a floor which measures 6 M. long and 5.25 M. broad? Ans. 31.5 M 2 . 17. A pavement measures 39.2 M. long and 4.15 M. broad, how many square M. does it contain? Ans. 162.68 M. a 18. A strip of land 120 M. long contains 342 ares, what is its breadth? Ans. 285 M. 19. What will 545.37 centares cost at $4.15 per are ? Ans. $22.63. 20. From 1374.27 M 2 . take 7.14 ares. Ans. 6.6027 M 2 . 21. At $50.50 per H. A., what will 1371.154 ares cost? Ans. $692.43. 22. In 37.27 M 3 . how many decisteres? How many dekasteres? Ans. 372.7 d. s., 3.727 D. S. 23. At $1.50 per M 3 ., what will 157.28 dekasteres cost? Ans. $2359.2. 294 NELSON'S COMMON-SCHOOL ARITHMETIC. 24. What will 153 S. of wood cost at $1.75 per S.? Ans. $267.75. 25. $875.55 was paid for 413.5 S. of wood, what was the cost per stere? Ans. $2.17. 26. A cistern is 3 M. long, 2 M. wide, and 1 M. deep, what weight of water will it contain? Ans. 13227.6 Ibs., or 6 tonneau. 27. What will a decistere of wood cost, at $3.57 per stere? Ans. 36 cents. 28. Find the price of 53 liters of wine, at $3.50 per dekaliter. Ans. $18.55. 29. A vessel measures 53.5 liters, what weight of water does it contain? Ans. 53.5 kilograms. 30. In .3514 H. L. how many liters? Ans. 35.14 L. 31. In 145.37 M 3 . how many liters? Ans. 145370 L. 32. Find the price of 5.3 liters of wine, at $1.75. Ans. $92.75: 33. At 57 cents a meter, what will 317.5 meters cost? Ans. $180.975. 34. An are of land cost $53, what was the price per meter? Ans. 53 cents. 35. A kilogram costs $37.50, what will a dekagram cost? Ans. 37-J- cents. 36. How many ares in a square piece of land which measures 150 meters in length and 500 in breadth? Ans. 750. 37. At $25 a kilogram, what will a dekagram cost? Ans. 25 cents. 38. At $7.25 a stere, what will 320 steres of wood cost? Ans. $2320. THE METRIC SYSTEM. 295 343. To reduce the denominations of the ordinary to those of the Metric System. 1. In 3 miles 35 rods, how many meters? 3 miles 35 rods=197010 inches, which, divided by 39.37, (the number of inches in a meter,) gives 5004.06 meters. 2. In 5^ yards how many meters? Ans. 4.877. 3. Reduce 37 |~ feet to meters. Ans. 11.43. 4. In 29 inches how many centimeters? Ans. 73.66. 5. In 6 inches how many millimeters? Aus. 152.4. 61 How many square meters in 57 rods? Ans. 1441.68. 7. In 5 acres how many ares? Ans. 202.34. 8. How many liters of wine in 37 gallons? Ans. 140.06. 9. In 57 yards of carpet how many meters? Ans. 52.12. 10. How many liters in 5^- bushels? Ans. 193.82. 11. How many cubic meters in 3759 cubic feet? Ans. 106.44 12. Reduce 3 tons to tonneaus. Ans. 2.72. 13. At $2.50 a liter, what will 20 gallons cost? Ans. $189.27 14. At 75 cents a meter, what will 135 yards cost? Ans. $92.58. 344. To reduce denominations of the Metric to those of the ordinary system. 1. In 175 meters how many yards? Ans. 191 yds., 1 ft., If in. In 1 meter there are 39.37 inches, in 175 meters there are 175 times as many. 39.37X175=6889.75 inches, which, reduced to yards, rr=191 yards, 1 foot, ly^ inches. 2. In 379.53 meters how many yards? Ans. 415.058. 296 NELSON'S COMMON-SCHOOL ARITHMETIC. 3. Reduce 743.5 K. M. to miles. Ans. 461.988 4. In 2435 millimeters how many inches? Ans. 95.866.. 5. Reduce 50 pounds to kilograms. Ans. 22.68. 6. At $2.75 per kilogram, what will 375 Ibs. cost? Ans. $467.77, 7. At 35 cents per lb., what is it per kilogram? Ans. 77 cents. 8. At $10.50 per cord, what is it per stere? Ans. $2.90. 9. At $1.25 a bushel, what should wheat be sold at per liter? Ans. 3^ cents. 10. A bushel of oats weighs 33 pounds, what should it weigh per liter? Ans. 1.14 Ibs. 11. How long will it take a man to travel a distance of 57 K. M. who walks at the rate of 3^ miles per hour? Ans. 10.119 hours. The following multipliers, taken from a pamphlet writ- ten by Prof. H. A. Newton, of Yale College, will facilitate the labor of converting the denominations of one system into those of the other. They will be found sufficiently correct for ordinary purposes : MetersX39.3685=inches. InchesX.0254=nneters. MetersX3.2807=feet. FeetX-30481 meters. MetersXl-09357=yards. YardsX.91444=meters. MetersX-19883r=:rods. RodsX5.0294=meters. KilometersX-62135=miles. MilesXl-6094=kilometers. Sq. metersX!550 sq. inches. Sq. inchesX.0006452=sq. M. Sq. metersX10.763=rrsq. feet. Sq. feetX-09291 sq. meters. Sq. metersXl-196 sq. yards. Sq. 3'ardsX.8362=sq. meters. AresX3.953=sq. rods. Sq. rodsX.2529^ares. HectaresX2.4709=racres. AcresX.4047=hectares. THE METRIC SYSTEM. 297 Hect.aresX-003861=sq. miles. Sq. milesX259.=rhectares. LitersX33.81 fluid ounces. Fluid ouncesX-02958 liters. LitersXl-05656=quarts. QuartsX-9465r=liters. LitersX.26414:=gallons. GallonsX3.786=liters. HectolitersX2.837=bushels. BushelsX.3524=hectoliters. LitersX61.012 cubic inches. Cubic inchesX.01639=rliters. HectolitersX3.531=cubic feet. Cubic feetX.2832=rhectoliters. SteresXl-3078 cubic yards. Cubic yardsX-7646 steres. SteresX-2759=cords. CordsX3.625z=Steres. GramsXl5-44 grains. GrainsX-0648=grams. KilogramsX32.147--=troy ounces. Troy oz.X-03108=rK. G. KilogramsX35.30=avoirdupois oz. Av. oz.X-02833^K. G. KilogramsX2.681=Troy pounds. Troy poundsX-373=K. G. KilogramsX2.206=Av. pounds. Av. poundsX4536=K. G. TonneausX-985=long tons. Long tonsXl-015=:tonneaus. TonneausXl-103=short tons. Short tonsX-9066 tonneaus. 1. In 37 meters how many yards? Ans. 37X1.09357=40.462 yds. 2. In 40.462 yards how many meters? Ans. 40.462 X. 91444=37 meters. 3. In 120 kilograms how many pounds Avoirdupois? Ans. 120X2.206=264.72 pounds. 4. In 26.472 pounds how many kilograms? Ans. 264.72X. 4536=120. kilograms. The meter is nearly 3 feet 3 inches and 3 eighths, or 3 feet 3J- inches. REMARKS. The word Meter means measure, as in gas- meter, or gas-measure. Stere means solid, as in stereotype, solid type, or rather a solid mass of type. Stereoscope, an instrument to make two pictures look like a solid. Are signifies area, a surface included within given lines. Quintal is from the Latin root centum, a hundred ; for- merly a hundred weight. PAGE. Vocabulary of Technical Terms used in Business 5 Tables of Money, Weights and Measures 15 Notation and Numeration 26 Addition...! 33 Subtraction 43 Multiplication 49 Division 60 Short Division 61 Percentage 68 Long Division 72 Properties of Numbers 77 Multiplication by Aliquots 79 Easy Fractions 82 The Mercantile Profession 94 Bills and Invoices 100 Compound Numbers 117 Short Methods 125 Marking Goods Gain and Loss 132 Commission and Brokerage 139 Interest and forms of notes, etc 142 Simple Interest ; 143 Banking 154 Checks, Drafts, etc 163 Receipts 165 Time Tables for Averaging.. 168 Discounting Notes % 170 True Discount 174 Compound Interest 178 Average Equation of Payments , 180 (298) NELSON'S COMMON-SCHOOL ARITHMETIC. 299 Accounts Current - 194 Exchange 196 Fractions. * 202 Extinction of Fractions 204 Decimals 206 Multiplication of Decimals v ... 210 Addition of Decimals 208 Subtraction of Decimals 209 Division of Decimals 211 Reduction of Decimals 213 Multiplication of Common Fractions 216 Subtraction of Common Fractions 222 Division of Common Fractions 219 Addition of Common Fractions... 223 Ratio and Proportion 227 Compound Proportion 231 Partnership 233 Joint Stock Companies, etc 235 Importing 239 Foreign Exchange 240 Insurance .-. 244 Duodecimals 246 Involution and Evolution 248 The Trades, Farming, etc 256 . Useful Hints on Building 257 Geometrical Definitions 278 Mensuration 280 Gauging 286 The Metric System, 287 - -2T YA 024 ID M3059S9 QA/ A/ THE UNIVERSITY OF CALIFORNIA LIBRARY \\ WHO Li BOOKSELLERS 86 STATIONERS, HAV A LARGE, "ENTIRELY NEW, AND WELL ASSORTED STOCK OF BOOKS AND STATIONERY, OF EVE]JY IMAGINABLE KIND, School Books of every ki"d used in the West, .i "iril I'll**, I Plain A Fancy Letter and Note Paper, Foreign and Domestic Stationery, l-oolscrp, and Fiat cap Paper, Gold J.Vw.s, /, v*f,n .-,1s. ,,./ Onilf / -/^ lintu-rtx. Pa JUT R. W. CARROLL <& CO.- Static xi" .'h Street,